https://peda.net/id/93efdace5fa 2019-04-15T20:47:46+03:00 https://peda.net/:static/532/peda.net.logo.bg.svg <div class="license">Tämän sivun lisenssi <a rel="license" href="https://peda.net/info">Peda.net-yleislisenssi</a></div> https://peda.net/id/97efd9287d7 2019-05-24T01:23:49+03:00 <span>K6</span><br/> <span>Autoon kohtistuvien voimien summa on tukivoiman ja painon summa </span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3D%5Coverline%7BN%7D%2B%5Coverline%7BG%7D" alt="\Sigma\overline{F}=\overline{N}+\overline{G}"/><span> </span><br/> <span>Koska auto on ilmassa, siihen ei enää vaikuttaa tukivoimaa, tällöin tukivoima </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BN%7D%3D0" alt="\overline{N}=0"/><span> </span><br/> <span>Newtonin II laen mukaan autoon kohtistuvien voimien summa on kappaleen massa ja kiihtyvyyden tull, eli</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D_n" alt="\Sigma\overline{F}=m\overline{a}_n"/><br/> <span>Valitaan kiihtyvyyden ja painovoiman suunta positiiviseksi, saadaan skalaari yhtälö</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=G%3Dma_n" alt="G=ma_n"/><br/> <span>koska </span><img src="https://math-demo.abitti.fi/math.svg?latex=G%3Dmg" alt="G=mg"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=mg%3Dma_n%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3Am" alt="mg=ma_n\ \ \ \ \ \left|\right|:m"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%3Da_n" alt="g=a_n"/><br/> <span>Kun </span><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="a_n=\frac{v^2}{r}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%3D%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="g=\frac{v^2}{r}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%3D9%7B%2C%7D81%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="g=9{,}81\ \frac{m}{s^2}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r%3D45m" alt="r=45m"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3D%3F" alt="v=?"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%3D%5Cfrac%7Bv%5E2%7D%7Br%7D%5C%20%5Cleftrightarrow%5C%20v%3D%5Csqrt%5B%5D%7Bgr%7D%3D%5Csqrt%5B%5D%7B9%7B%2C%7D81%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%5Ccdot45m%7D%3D21%7B%2C%7D0107...%5Capprox21%7B%2C%7D01%5Cfrac%7Bm%7D%7Bs%7D" alt="g=\frac{v^2}{r}\ \leftrightarrow\ v=\sqrt[]{gr}=\sqrt[]{9{,}81\ \frac{m}{s^2}\cdot45m}=21{,}0107...\approx21{,}01\frac{m}{s}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=21%7B%2C%7D01%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Ccdot3%7B%2C%7D6%3D75%7B%2C%7D636%5C%20%5Cfrac%7Bkm%7D%7Bh%7D%5Capprox76%5C%20%5Cfrac%7Bkm%7D%7Bh%7D" alt="21{,}01\ \frac{m}{s}\cdot3{,}6=75{,}636\ \frac{km}{h}\approx76\ \frac{km}{h}"/><br/> <br/> K9 <div>a)</div> <div>Voidaan käyttää Newtonin gravitaatiovoiman kaava</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%3D%5C%20%5Cgamma%5Cfrac%7Bm_1m_2%7D%7Br%5E2%7D%3D6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot%5Cfrac%7B1200kg%5Ccdot1600kg%7D%7B2%7B%2C%7D5%5E2m%7D%3D2%7B%2C%7D05033816%5Ccdot10%5E%7B-5%7D%5C%20N%5Capprox2%7B%2C%7D1%5Ccdot10%5E%7B-5%7DN" alt="F=\ \gamma\frac{m_1m_2}{r^2}=6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}\cdot\frac{1200kg\cdot1600kg}{2{,}5^2m}=2{,}05033816\cdot10^{-5}\ N\approx2{,}1\cdot10^{-5}N"/></div> <div>b)</div> Newtonin toisen lain mukaan <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D_n" alt="\Sigma\overline{F}=m\overline{a}_n"/>. Koska gravitaatiovoiman suuruus on <img src="https://math-demo.abitti.fi/math.svg?latex=F%3D%5Cgamma%5Cfrac%7Bm_1m_2%7D%7Br%5E2%7D" alt="F=\gamma\frac{m_1m_2}{r^2}"/>,<span> </span>normaalikiihtyvyyden suurus on<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D_n%3Dm%5Ccdot%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="\overline{a}_n=m\cdot\frac{v^2}{r}"/> ja ilmanvastus pieni, saadaan yhtälö <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5Cfrac%7BmM%7D%7B%5Cleft(2R%5Cright)%5E2%7D%3Dm%5Cfrac%7Bv%5E2%7D%7B2R%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3Am" alt="\gamma\frac{mM}{\left(2R\right)^2}=m\frac{v^2}{2R}\ \ \ \ \ \left|\right|:m"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5Cfrac%7BM%7D%7B%5Cleft(2R%5Cright)%5E2%7D%3D%5Cfrac%7Bv%5E2%7D%7B2R%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Ccdot2R" alt="\gamma\frac{M}{\left(2R\right)^2}=\frac{v^2}{2R}\ \ \ \ \ \left|\right|\cdot2R"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5Cfrac%7BM%7D%7B2R%7D%3Dv%5E2%5C%20" alt="\gamma\frac{M}{2R}=v^2\ "/>, jossa R on Maan säde<br/> <div>Nopeus on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5Cfrac%7BM%7D%7B2R%7D%3Dv%5E2%5C%20%5C%20%5Cleftrightarrow%5C%20v%3D%5Csqrt%5B%5D%7B%5Cfrac%7B%5Cgamma%20M%7D%7B2R%7D%7D%3D%5Csqrt%5B%5D%7B%5Cfrac%7B6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%7D%7B2%5Ccdot6378.140%5Ccdot10%5E6m%7D%3D%7D5590%7B%2C%7D833...%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Capprox5%7B%2C%7D59%5C%20%5Cfrac%7Bkm%7D%7Bs%7D" alt="\gamma\frac{M}{2R}=v^2\ \ \leftrightarrow\ v=\sqrt[]{\frac{\gamma M}{2R}}=\sqrt[]{\frac{6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}\cdot5{,}974\cdot10^{24}kg}{2\cdot6378.140\cdot10^6m}=}5590{,}833...\ \frac{m}{s}\approx5{,}59\ \frac{km}{s}"/><br/> <br/> K11 <div>Gravitaatiokent'n voimakkuus saadaan kaavasta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g%5Cmathrm%7B_%7BPlaneetta%7D%7D%3D%5Cfrac%7B%5Cgamma%20m%5Cmathrm%7B_%7BPlaneetta%7D%7D%7D%7Br_%7B%5Cmathrm%7BPlaneetta%7D%7D%5E2%7D%3D%5Cfrac%7B%5Cgamma100M%7D%7B%5Cleft(10R%5Cright)%5E2%7D%3D%5Cfrac%7B%5Cgamma100M%7D%7B100R%5E2%7D%3D%5Cfrac%7B%5Cgamma%20M%7D%7BR%5E2%7D%3D%5Cfrac%7B6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%7D%7B%5Cleft(6378%7B%2C%7D140%5Ccdot10%5E3m%5Cright)%5E2%7D%3D9%7B%2C%7D8012...%5Capprox9.80%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="g\mathrm{_{Planeetta}}=\frac{\gamma m\mathrm{_{Planeetta}}}{r_{\mathrm{Planeetta}}^2}=\frac{\gamma100M}{\left(10R\right)^2}=\frac{\gamma100M}{100R^2}=\frac{\gamma M}{R^2}=\frac{6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}\cdot5{,}974\cdot10^{24}kg}{\left(6378{,}140\cdot10^3m\right)^2}=9{,}8012...\approx9.80\ \frac{m}{s^2}"/></div> </div> <div> <div>K12</div> </div> <div>Newtonin II lain mukaan <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D_n" alt="\Sigma\overline{F}=m\overline{a}_n"/>, koska gravitaatiovoiman suuruus on <img src="https://math-demo.abitti.fi/math.svg?latex=F%3D%5Cgamma%5Cfrac%7Bm_1m_2%7D%7Br%5E2%7D" alt="F=\gamma\frac{m_1m_2}{r^2}"/> , normaalikiihtyvyyden suuruus on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D_n%3D%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="\overline{a}_n=\frac{v^2}{r}"/> ja ilmavastus on pieni, voidaan muodosta yhtälö</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5Cfrac%7BmM%7D%7Br%5E2%7D%3Dm%5Cfrac%7Bv%5E2%7D%7Br%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3Am" alt="\gamma\frac{mM}{r^2}=m\frac{v^2}{r}\ \ \ \ \ \left|\right|:m"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cgamma%20M%7D%7Br%5E2%7D%3D%5Cfrac%7Bv%5E2%7D%7Br%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Ccdot%20r" alt="\frac{\gamma M}{r^2}=\frac{v^2}{r}\ \ \ \ \ \left|\right|\cdot r"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cgamma%20M%7D%7Br%7D%3Dv%5E2%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Ccdot%20r" alt="\frac{\gamma M}{r}=v^2\ \ \ \ \ \left|\right|\cdot r"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%20M%3Drv%5E2%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A%5Cgamma" alt="\gamma M=rv^2\ \ \ \ \ \left|\right|:\gamma"/><span>, jossa M on Auringon massa</span> <div> Auringon massa on<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=M%3D%5Cfrac%7Brv%5E2%7D%7B%5Cgamma%7D%3D%5Cfrac%7B227%7B%2C%7D9%5Ccdot10%5E9m%5Ccdot%5Cleft(24130%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Cright)%5E2%7D%7B6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%7D%3D1%7B%2C%7D98818...%5Ccdot10%5E%7B30%7Dkg%5Capprox1%7B%2C%7D988%5Ccdot10%5E%7B30%7Dkg" alt="M=\frac{rv^2}{\gamma}=\frac{227{,}9\cdot10^9m\cdot\left(24130\ \frac{m}{s}\right)^2}{6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}}=1{,}98818...\cdot10^{30}kg\approx1{,}988\cdot10^{30}kg"/></div> K15<br/> <span>a)</span> <div>Jaksoaika:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=T%3D%5Cfrac%7B32%7D%7B13%7Ds%3D2%7B%2C%7D461538462s%5Capprox2%7B%2C%7D5s" alt="T=\frac{32}{13}s=2{,}461538462s\approx2{,}5s"/></div> <div>Taajuus:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%3D%5Cfrac%7B1%7D%7B2%7B%2C%7D461538462s%7D%3D0%7B%2C%7D40625%5Capprox0%7B%2C%7D41%5Cmathrm%7BHz%7D" alt="f=\frac{1}{2{,}461538462s}=0{,}40625\approx0{,}41\mathrm{Hz}"/></div> <span>b)</span> <div> Jaksoaika:<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=T%3D%5Cfrac%7B60%7D%7B157%7Ds%3D0%7B%2C%7D3821656051s%5Capprox0%7B%2C%7D38s" alt="T=\frac{60}{157}s=0{,}3821656051s\approx0{,}38s"/><br/> Taajuus:<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%3D%5Cfrac%7B1%7D%7B0%7B%2C%7D3821656051s%7D%3D2%7B%2C%7D616...%5Capprox2%7B%2C%7D6%5Cmathrm%7BHz%7D" alt="f=\frac{1}{0{,}3821656051s}=2{,}616...\approx2{,}6\mathrm{Hz}"/></div> <br/> <div><span> </span></div> </div> 2019-05-23T19:27:01+03:00 https://peda.net/id/d731b91e7c5 2019-05-22T09:50:09+03:00 <span>15-2</span> <div> Ihmisen kuuloalue kuvaajan mukaan 100 dB:n intensitettitasoa vastaa intensiteetti </div> <img src="https://math-demo.abitti.fi/math.svg?latex=I%3D10%5E%7B-6%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D" alt="I=10^{-6}\ \frac{W}{m^2}"/> <div>Äänen teho saadan kaavalla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=I%3D%5Cfrac%7BP%7D%7BA%7D%5C%20%5Cleftrightarrow%5C%20P%3DIA%3D10%5E%7B-6%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%5Ccdot%5Cpi%5Ccdot%5Cleft(%5Cfrac%7B0%7B%2C%7D00844m%7D%7B2%7D%5Cright)%5E2%3D5%7B%2C%7D5417...%5Ccdot10%5E%7B-11%7D%3D5%7B%2C%7D5%5Ccdot10%5E%7B-11%7DW" alt="I=\frac{P}{A}\ \leftrightarrow\ P=IA=10^{-6}\ \frac{W}{m^2}\cdot\pi\cdot\left(\frac{0{,}00844m}{2}\right)^2=5{,}5417...\cdot10^{-11}=5{,}5\cdot10^{-11}W"/><br/> <div> <br/> <div>15-4</div> <div>Jos ääni etenee vapaasti kaikkii suuntiin, on kahdella etäisyydellä<img src="https://math-demo.abitti.fi/math.svg?latex=r_1" alt="r_1"/>ja<img src="https://math-demo.abitti.fi/math.svg?latex=r_2" alt="r_2"/> mitattujen intensiteettien suhde on </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7BI_1%7D%7BI_2%7D%3D%5Cfrac%7Br_2%5E2%7D%7Br_1%5E2%7D%5C%20" alt="\frac{I_1}{I_2}=\frac{r_2^2}{r_1^2}\ "/></div> <div>Tehtävässä pitää laskea mikä on se toinen etäisyys, eli <img src="https://math-demo.abitti.fi/math.svg?latex=r_2" alt="r_2"/>, joten muutetaan alkuperäisen kaavan tehää muotoon<br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7BI_1%7D%7BI_2%7D%3D%5Cfrac%7Br_2%5E2%7D%7Br_1%5E2%7D%5C%20%5C%20%5Cleftrightarrow%5C%20r_2%3D%5Csqrt%5B%5D%7B%5Cleft(r_1%5E2%5Ccdot%5Cfrac%7BI_1%7D%7BI_2%7D%5Cright)%7D%3D%5Csqrt%5B%5D%7B12%5E2m%5Ccdot%5C%20%5Cfrac%7B10%5E%7B-4%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%7B1%7B%2C%7D0%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%7D%3D0%7B%2C%7D12m" alt="\frac{I_1}{I_2}=\frac{r_2^2}{r_1^2}\ \ \leftrightarrow\ r_2=\sqrt[]{\left(r_1^2\cdot\frac{I_1}{I_2}\right)}=\sqrt[]{12^2m\cdot\ \frac{10^{-4}\ \frac{W}{m^2}}{1{,}0\ \frac{W}{m^2}}}=0{,}12m"/></div> <div> </div> <div>15-6</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=I_0%3D10%5E%7B-12%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D" alt="I_0=10^{-12}\ \frac{W}{m^2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=10I_0%3D10%5Ccdot10%5E%7B-12%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%3D10%5E%7B-11%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D" alt="10I_0=10\cdot10^{-12}\ \frac{W}{m^2}=10^{-11}\ \frac{W}{m^2}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=100I_0%3D10%5E2I_0%3D10%5E2%5Ccdot10%5E%7B-12%7D%5Cfrac%7BW%7D%7Bm%5E2%7D%3D10%5E%7B-10%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D" alt="100I_0=10^2I_0=10^2\cdot10^{-12}\frac{W}{m^2}=10^{-10}\ \frac{W}{m^2}"/></div> <div>Laketaan äänen intensiteettitasoa kaavalla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=L%3D10dB%5Ccdot%5Clog%5Cfrac%7BI%7D%7BI_0%7D" alt="L=10dB\cdot\log\frac{I}{I_0}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=L_1%3D10dB%5Ccdot%5Clog%5Cfrac%7B10%5E%7B-12%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%7B10%5E%7B-12%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%3D10dB%5Ccdot0%3D0dB" alt="L_1=10dB\cdot\log\frac{10^{-12}\ \frac{W}{m^2}}{10^{-12}\ \frac{W}{m^2}}=10dB\cdot0=0dB"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=L_2%3D10dB%5Ccdot%5Clog%5Cfrac%7B10%5E%7B-11%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%7B10%5E%7B-12%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%3D10dB%5Ccdot1%3D10dB" alt="L_2=10dB\cdot\log\frac{10^{-11}\ \frac{W}{m^2}}{10^{-12}\ \frac{W}{m^2}}=10dB\cdot1=10dB"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=L_3%3D10dB%5Ccdot%5Clog%5Cfrac%7B10%5E%7B-10%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%7B10%5E%7B-12%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%3D10dB%5Ccdot2%3D20dB" alt="L_3=10dB\cdot\log\frac{10^{-10}\ \frac{W}{m^2}}{10^{-12}\ \frac{W}{m^2}}=10dB\cdot2=20dB"/> <div> </div> <div> <div>15-7  <div>(MAOL s.91) Ihmisen kuuloalue kuvaajan mukaan 100 dB:n intensitettitasoa vastaa intensiteetti </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=I%3D10%5E%7B-2%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D" alt="I=10^{-2}\ \frac{W}{m^2}"/></div> Kymmenen lehtipuhaltimen äänen intensiteettitaso on  <div><img src="https://math-demo.abitti.fi/math.svg?latex=I_%7B10%7D%3D10%5Ccdot10%5E%7B-2%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%3D10%5E%7B-1%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D" alt="I_{10}=10\cdot10^{-2}\ \frac{W}{m^2}=10^{-1}\ \frac{W}{m^2}"/></div> jaintensiteettitaso on <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=L%3D10dB%5Ccdot%5Clog%5Cfrac%7BI_%7B10%7D%7D%7BI_0%7D%3D10dB%5Ccdot%5Clog%5Cfrac%7B10%5E%7B-1%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%7B10%5E%7B-12%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%3D10dB%5Ccdot%5Clog10%5E%7B11%7D%3D10dB%5Ccdot11%3D110dB" alt="L=10dB\cdot\log\frac{I_{10}}{I_0}=10dB\cdot\log\frac{10^{-1}\ \frac{W}{m^2}}{10^{-12}\ \frac{W}{m^2}}=10dB\cdot\log10^{11}=10dB\cdot11=110dB"/></div> </div> <div> </div> <div>15-9</div> </div> </div> <div> (MAOL s.91) Ihmisen kuuloalue kuvaajan mukaan 120 dB:n intensiteettitasoa vastaa intensiteetti</div> <img src="https://math-demo.abitti.fi/math.svg?latex=I_1%3D10%5E0%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D" alt="I_1=10^0\ \frac{W}{m^2}"/><br/> <div>ja 130 Hz:n intensiteettitasoa vastaa intensiteetti  </div> <img src="https://math-demo.abitti.fi/math.svg?latex=I_2%5Capprox10%5E%7B-9%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D" alt="I_2\approx10^{-9}\ \frac{W}{m^2}"/> <div> Jos ääni etenee vapaasti kaikkii suuntiin, on kahdella etäisyydellä<img src="https://math-demo.abitti.fi/math.svg?latex=r_1" alt="r_1"/>ja<img src="https://math-demo.abitti.fi/math.svg?latex=r_2" alt="r_2"/> mitattujen intensiteettien suhde on </div> <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7BI_1%7D%7BI_2%7D%3D%5Cfrac%7Br_2%5E2%7D%7Br_1%5E2%7D%5C%20" alt="\frac{I_1}{I_2}=\frac{r_2^2}{r_1^2}\ "/><br/> Tehtävässä pitää laskea mikä on se toinen etäisyys, eli <img src="https://math-demo.abitti.fi/math.svg?latex=r_2" alt="r_2"/>, joten muutetaan alkuperäisen kaavan tehää muotoon<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7BI_1%7D%7BI_2%7D%3D%5Cfrac%7Br_2%5E2%7D%7Br_1%5E2%7D%5C%20%5Cleftrightarrow%5C%20r_2%3D%5Csqrt%5B%5D%7B%5Cleft(r_1%5E2%5Ccdot%5Cfrac%7BI_2%7D%7BI_1%7D%5Cright)%7D%3D%5Csqrt%5B%5D%7B1%5E2m%5Ccdot%5Cfrac%7B10%5E0%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%7B10%5E%7B-9%7D%5C%20%5Cfrac%7BW%7D%7Bm%5E2%7D%7D%7D%3D31622%7B%2C%7D7766m%3D31%7B%2C%7D6227766km%5Capprox32km" alt="\frac{I_1}{I_2}=\frac{r_2^2}{r_1^2}\ \leftrightarrow\ r_2=\sqrt[]{\left(r_1^2\cdot\frac{I_2}{I_1}\right)}=\sqrt[]{1^2m\cdot\frac{10^0\ \frac{W}{m^2}}{10^{-9}\ \frac{W}{m^2}}}=31622{,}7766m=31{,}6227766km\approx32km"/><br/> <br/> <div>  </div> </div> </div> 2019-05-22T09:50:09+03:00 https://peda.net/id/0aceae5c7b1 2019-05-20T20:04:35+03:00 <div> <div>14-1</div> </div> <div>Huilun sävel riippuu siitä ulottuvan äänen taajuudesta</div> <div>Taajuutta voidaan saada kaavalla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D" alt="v=f\lambda\ \leftrightarrow\ f=\frac{v}{\lambda}"/></div> <div>Poikkihuiluun syntyvän perussävelen taajuudeksi saadaan</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D%3D%5Cfrac%7Bv%7D%7B2L%7D" alt="f=\frac{v}{\lambda}=\frac{v}{2L}"/></div> <span>Klarinettiin syntyvän perussävelen taajudeksi saadaan</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D%3D%5Cfrac%7Bv%7D%7B4L%7D" alt="f=\frac{v}{\lambda}=\frac{v}{4L}"/> <div>Koska poikkihuiluun syntyvän perussävelen taajuus on pienempi, tällöin ääniaallot ovat asetettu tiheämmin, näin ollen poikkihuiluun syntyvän perussävelen on korkeampi.</div> <div><br/> <div>14-3</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3D343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Cleft(%5Cmathrm%7B%C3%A4%C3%A4nennopeus%5C%20huonel%C3%A4mm%C3%B6ss%C3%A4%5C%20%7D%5Cright)" alt="v=343\ \frac{m}{s}\left(\mathrm{äänennopeus\ huonelämmössä\ }\right)"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%3D69%7B%2C%7D3Hz" alt="f=69{,}3Hz"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clambda%3D%3F" alt="\lambda=?"/><br/> Aallonpituutta voidaan laskea kaavalla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D%5Cfrac%7Bv%7D%7Bf%7D%3D%5Cfrac%7B343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B69%7B%2C%7D3%5C%20Hz%7D%3D4%7B%2C%7D94949...%5Capprox4%7B%2C%7D9495m" alt="v=f\lambda\ \leftrightarrow\ \lambda=\frac{v}{f}=\frac{343\ \frac{m}{s}}{69{,}3\ Hz}=4{,}94949...\approx4{,}9495m"/><span> </span></div> <div>Koska bassoklarinetti on puoleksi suljettu putki, sen pituus on lambdan kanssa suhteessa</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=L%3D%5Cfrac%7B1%7D%7B4%7D%5Clambda%5C%20%5Cleftrightarrow%5C%20L%3D%5Cfrac%7B%5Clambda%7D%7B4%7D%3D%5Cfrac%7B4%7B%2C%7D9495%7D%7B4%7D%3D1%7B%2C%7D237...%5Capprox1%7B%2C%7D24m" alt="L=\frac{1}{4}\lambda\ \leftrightarrow\ L=\frac{\lambda}{4}=\frac{4{,}9495}{4}=1{,}237...\approx1{,}24m"/></div> <div><br/> <div>14-4</div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=L%3D124cm%3D1%7B%2C%7D24m" alt="L=124cm=1{,}24m"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3D343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Cleft(%5Cmathrm%7B%C3%A4%C3%A4nennopeus%5C%20huonel%C3%A4mm%C3%B6ss%C3%A4%5C%20%7D%5Cright)" alt="v=343\ \frac{m}{s}\left(\mathrm{äänennopeus\ huonelämmössä\ }\right)"/><span> </span> <div>a)</div> <div>Perussävel:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=L%3D%5Cfrac%7B%5Clambda%7D%7B2%7D%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D2L%3D2%5Ccdot1%7B%2C%7D24m%3D2%7B%2C%7D48m" alt="L=\frac{\lambda}{2}\ \leftrightarrow\ \lambda=2L=2\cdot1{,}24m=2{,}48m"/><span> </span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D%3D%5Cfrac%7B343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B2%7B%2C%7D48%7D%3D138%7B%2C%7D3...%5Capprox138%5C%20Hz" alt="v=f\lambda\ \leftrightarrow\ f=\frac{v}{\lambda}=\frac{343\ \frac{m}{s}}{2{,}48}=138{,}3...\approx138\ Hz"/></div> <div>Ensimmäinen yläsävel:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=L%3D%5Clambda%5C%20%3D1%7B%2C%7D24m" alt="L=\lambda\ =1{,}24m"/><span>  </span></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D%3D%5Cfrac%7B343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B1%7B%2C%7D24%7D%3D276%7B%2C%7D6...%5Capprox277Hz" alt="v=f\lambda\ \leftrightarrow\ f=\frac{v}{\lambda}=\frac{343\ \frac{m}{s}}{1{,}24}=276{,}6...\approx277Hz"/> </div> <div><span>b)</span> <div>Perussävel: <br/> Jos on toisesta päästä suljettu putki, aallonpituus olisi tällöin <div><img src="https://math-demo.abitti.fi/math.svg?latex=L%3D%5Cfrac%7B%5Clambda%7D%7B4%7D%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D4L%3D4%5Ccdot1%7B%2C%7D24m%3D4.96m" alt="L=\frac{\lambda}{4}\ \leftrightarrow\ \lambda=4L=4\cdot1{,}24m=4.96m"/><span>  </span></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D%3D%5Cfrac%7B343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B4%7B%2C%7D96%7D%3D69%7B%2C%7D15...%5Capprox69%7B%2C%7D2Hz" alt="v=f\lambda\ \leftrightarrow\ f=\frac{v}{\lambda}=\frac{343\ \frac{m}{s}}{4{,}96}=69{,}15...\approx69{,}2Hz"/> </div> <div>Ensimmäinen yläsävel:</div> <div>Jos on toisesta päästä suljettu putki, aallonpituus olisi tällöin <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=L%3D%5Cfrac%7B3%5Clambda%7D%7B4%7D%5Cleftrightarrow%5C%20%5Clambda%3D%5Cfrac%7B4L%7D%7B3%7D%3D%5Cfrac%7B4%5Ccdot1%7B%2C%7D24m%7D%7B3%7D%3D1%7B%2C%7D65333....%5Capprox1%7B%2C%7D6533m" alt="L=\frac{3\lambda}{4}\leftrightarrow\ \lambda=\frac{4L}{3}=\frac{4\cdot1{,}24m}{3}=1{,}65333....\approx1{,}6533m"/><span> </span></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D%3D%5Cfrac%7B343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B1%7B%2C%7D6533%7D%3D207%7B%2C%7D4...%5Capprox207Hz" alt="v=f\lambda\ \leftrightarrow\ f=\frac{v}{\lambda}=\frac{343\ \frac{m}{s}}{1{,}6533}=207{,}4...\approx207Hz"/></div> <div><br/> <div>14-5</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=440%7B%2C%7D0Hz%5Cpm2%7B%2C%7D00Hz" alt="440{,}0Hz\pm2{,}00Hz"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%3D442%7B%2C%7D0Hz" alt="=442{,}0Hz"/></div> <div>tai</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%3D438%7B%2C%7D0Hz" alt="=438{,}0Hz"/></div> <br/> <div>14-8</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3D340%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=340\ \frac{m}{s}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=L%3D0%7B%2C%7D60m" alt="L=0{,}60m"/></div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=L%3D0%7B%2C%7D60m%3D%5Cfrac%7B%5Clambda%7D%7B4%7D%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D4L%3D4%5Ccdot0%7B%2C%7D60m%3D2%7B%2C%7D40m" alt="L=0{,}60m=\frac{\lambda}{4}\ \leftrightarrow\ \lambda=4L=4\cdot0{,}60m=2{,}40m"/><span> </span></div> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D%3D%5Cfrac%7B340%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B2%7B%2C%7D40%7D%3D141%7B%2C%7D66...%5Capprox142Hz" alt="v=f\lambda\ \leftrightarrow\ f=\frac{v}{\lambda}=\frac{340\ \frac{m}{s}}{2{,}40}=141{,}66...\approx142Hz"/></div> <div>b)</div> </div> <div>Jos pillissä on ilman sijasta heliumia, äänen nopeus olisi tällöin </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3D965%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=965\ \frac{m}{s}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=L%3D0%7B%2C%7D60m%3D%5Cfrac%7B%5Clambda%7D%7B4%7D%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D4L%3D4%5Ccdot0%7B%2C%7D60m%3D2%7B%2C%7D40m" alt="L=0{,}60m=\frac{\lambda}{4}\ \leftrightarrow\ \lambda=4L=4\cdot0{,}60m=2{,}40m"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D%3D%5Cfrac%7B965%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B2%7B%2C%7D40%7D%3D402%7B%2C%7D08...%5Capprox402Hz" alt="v=f\lambda\ \leftrightarrow\ f=\frac{v}{\lambda}=\frac{965\ \frac{m}{s}}{2{,}40}=402{,}08...\approx402Hz"/></div> 2019-05-20T19:12:19+03:00 https://peda.net/id/17b1e2e877c 2019-05-16T14:16:46+03:00 <span>13-3</span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20%5Clambda_1%3D%5Cfrac%7Bv_1%7D%7Bf%7D%3D%5Cfrac%7B340%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B1%5C%20000Hz%7D%3D0%7B%2C%7D34m" alt="v=f\lambda\ \leftrightarrow\ \lambda_1=\frac{v_1}{f}=\frac{340\ \frac{m}{s}}{1\ 000Hz}=0{,}34m"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clambda_2%3D%5Cfrac%7Bv_2%7D%7Bf%7D%3D%5Cfrac%7B260%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B1%5C%20000Hz%7D%3D0%7B%2C%7D26m" alt="\lambda_2=\frac{v_2}{f}=\frac{260\ \frac{m}{s}}{1\ 000Hz}=0{,}26m"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csin%5Calpha_1%7D%7B%5Csin%5Calpha_2%7D%3D%5Cfrac%7B%5Clambda_1%7D%7B%5Clambda_2%7D%3D%5Cfrac%7B0%7B%2C%7D34m%7D%7B0%7B%2C%7D26m%7D%3D%5Cfrac%7B17%7D%7B13%7D%5Capprox1%7B%2C%7D3" alt="\frac{\sin\alpha_1}{\sin\alpha_2}=\frac{\lambda_1}{\lambda_2}=\frac{0{,}34m}{0{,}26m}=\frac{17}{13}\approx1{,}3"/><br/> c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csin%5Calpha_1%7D%7B%5Csin%5Calpha_2%7D%3D%5Cfrac%7B%5Clambda_1%7D%7B%5Clambda_2%7D%5C%20%5Cleftrightarrow%5C%20%5Csin%5Calpha_2%3D%5Cfrac%7B%5Csin%5Calpha_1%5Ccdot%5Clambda_2%7D%7B%5Clambda_1%7D%3D%5Cfrac%7B%5Csin45%C2%B0%5Ccdot0%7B%2C%7D26m%7D%7B0%7B%2C%7D34m%7D%3D0%7B%2C%7D54072...%5Capprox0%7B%2C%7D5407" alt="\frac{\sin\alpha_1}{\sin\alpha_2}=\frac{\lambda_1}{\lambda_2}\ \leftrightarrow\ \sin\alpha_2=\frac{\sin\alpha_1\cdot\lambda_2}{\lambda_1}=\frac{\sin45°\cdot0{,}26m}{0{,}34m}=0{,}54072...\approx0{,}5407"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5E%7B-1%7D%5Cleft(0%7B%2C%7D5407%5Cright)%3D32%7B%2C%7D731...%5Capprox32%7B%2C%7D73%C2%B0" alt="\sin^{-1}\left(0{,}5407\right)=32{,}731...\approx32{,}73°"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=45%C2%B0-32%7B%2C%7D73%C2%B0%3D12%7B%2C%7D27%C2%B0%5Capprox12%C2%B0" alt="45°-32{,}73°=12{,}27°\approx12°"/></div> <div><br/> <div>13-6</div> <div> a) Sukellusveneen potkuri lähettää koko ajan 45 Hz:n ääntä, joten äänen taajuus ei muutu. Aaltoliikkeen perusyhtälö on v = fλ. Koska äänen nopeus v pienenee ja taajuus f ei muutu rajapinnan ylittämisen jälkeen, ääniaallon aallonpituuden λ täytyy pienentyä.</div> <div>b)</div> <div> Koska äänen nopeus pienenee rajapinnan ylityksen jälkeen, ääniaaltojen etenemissuunta kääntyy normaaliin päin. Siten kokonaisheijastuminen ei ole mahdollinen. Näin ääniaallolle ovat mahdollisia kaikki tulokulman α1 arvot 0 ≤ α1 &lt; 90°, ja ääni läpäisee veden ja ilman rajapinnan. </div> <div> </div> <div>13-7</div> <div>Äänen nopeus 20°C:ssa vedessä on 1484 m/s ja 343 m/s ilmassa (Maol)</div> </div> <div>Nopeuden kaavan mukaan </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3D%5Cfrac%7Bs%7D%7Bt%7D%5C%20%5Cleftrightarrow%5C%20s%3Dvt%3D1484%5C%20%5Cfrac%7Bm%7D%7Bs%5C%20%7D%5Ccdot%20t" alt="v=\frac{s}{t}\ \leftrightarrow\ s=vt=1484\ \frac{m}{s\ }\cdot t"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=s%3D343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5C%20%5Ccdot%5Cleft(t%2B8%7B%2C%7D0s%5Cright)" alt="s=343\ \frac{m}{s}\ \cdot\left(t+8{,}0s\right)"/> <div>Koska etäisyys on sama, voidaan muodosta yhtälön</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1484%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Ccdot%20t%3D343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Ccdot%5Cleft(t%2B8%7B%2C%7D0s%5Cright)" alt="1484\ \frac{m}{s}\cdot t=343\ \frac{m}{s}\cdot\left(t+8{,}0s\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=t%3Dt%3D2.4049...%5Capprox2%7B%2C%7D405s" alt="t=t=2.4049...\approx2{,}405s"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%3D1484%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Ccdot2%7B%2C%7D405s%3D3569%7B%2C%7D02%5Capprox3600m%3D3%7B%2C%7D6km" alt="s=1484\ \frac{m}{s}\cdot2{,}405s=3569{,}02\approx3600m=3{,}6km"/></div> <div><br/> <div>13-9</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%3Df_0%5Ccdot%5Cfrac%7Bv%7D%7Bv-v_1%7D%3D405%5C%20Hz%5Ccdot%5Cfrac%7B343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B343%5C%20%5Cfrac%7Bm%7D%7Bs%7D-%5Cfrac%7B170%5C%20%5Cfrac%7Bkm%7D%7Bh%7D%7D%7B3%7B%2C%7D6%7D%7D%3D469%7B%2C%7D66...%5Capprox470Hz" alt="f=f_0\cdot\frac{v}{v-v_1}=405\ Hz\cdot\frac{343\ \frac{m}{s}}{343\ \frac{m}{s}-\frac{170\ \frac{km}{h}}{3{,}6}}=469{,}66...\approx470Hz"/></div> b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%3Df_0%5Ccdot%5Cfrac%7Bv%7D%7Bv-v_1%7D%3D405%5C%20Hz%5Ccdot%5Cfrac%7B343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B343%5C%20%5Cfrac%7Bm%7D%7Bs%7D%2B%5Cfrac%7B170%5C%20%5Cfrac%7Bkm%7D%7Bh%7D%7D%7B3%7B%2C%7D6%7D%7D%3D355%7B%2C%7D9894...%5Capprox360Hz" alt="f=f_0\cdot\frac{v}{v-v_1}=405\ Hz\cdot\frac{343\ \frac{m}{s}}{343\ \frac{m}{s}+\frac{170\ \frac{km}{h}}{3{,}6}}=355{,}9894...\approx360Hz"/><br/> 13-10</div> </div> 2019-05-16T14:16:46+03:00 https://peda.net/id/b91ebfe277b 2019-05-20T20:02:18+03:00 <div>12-2</div> <div>a) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%3D4%7B%2C%7D0Hz" alt="f=4{,}0Hz"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3D6%7B%2C%7D4%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=6{,}4\ \frac{m}{s}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=l%3D4%7B%2C%7D0m" alt="l=4{,}0m"/> <div>a)</div> <div>Lasketaan ensi mikä on aallonpituus</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D%5Cfrac%7Bv%7D%7Bf%7D%3D%5Cfrac%7B6%7B%2C%7D4%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B4%7B%2C%7D0%5C%20Hz%7D%3D1%7B%2C%7D6%5C%20m" alt="v=f\lambda\ \leftrightarrow\ \lambda=\frac{v}{f}=\frac{6{,}4\ \frac{m}{s}}{4{,}0\ Hz}=1{,}6\ m"/></div> <div>Yhden kuvun pituus on yhtä kuin aallonpituuden puolikas, joten:</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4%7B%2C%7D0%5C%20m%7D%7B%5Cfrac%7B1%7B%2C%7D6%5C%20m%7D%7B2%7D%7D%3D5" alt="\frac{4{,}0\ m}{\frac{1{,}6\ m}{2}}=5"/></div> <div>Kupuja on 5</div> <div>b)</div> <div>Jos taajust olisi 3,2 Hz, uusi aallonpituus olisi</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D%5Cfrac%7Bv%7D%7Bf%7D%3D%5Cfrac%7B6%7B%2C%7D4%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B3%7B%2C%7D2%5C%20Hz%7D%3D2%7B%2C%7D0%5C%20m" alt="v=f\lambda\ \leftrightarrow\ \lambda=\frac{v}{f}=\frac{6{,}4\ \frac{m}{s}}{3{,}2\ Hz}=2{,}0\ m"/> </div> <div>ja yhden kuvun pituus olisi</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B4%7B%2C%7D0%5C%20m%7D%7B%5Cfrac%7B2%7B%2C%7D0%5C%20m%7D%7B2%7D%7D%3D4" alt="\frac{4{,}0\ m}{\frac{2{,}0\ m}{2}}=4"/></div> <div>Kupuja on 4</div> <div><br/> <div>12-3</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox55cm%5Ccdot2%3D110cm" alt="\approx55cm\cdot2=110cm"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=l%3D%5Cmathrm%7BKuvun%5C%20m%C3%A4%C3%A4r%C3%A4%7D%5Ccdot%5Cfrac%7B%5Clambda%7D%7B2%7D%3D2%5Ccdot%5Cfrac%7B110%5C%20cm%7D%7B2%7D%3D110cm" alt="l=\mathrm{Kuvun\ määrä}\cdot\frac{\lambda}{2}=2\cdot\frac{110\ cm}{2}=110cm"/></div> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%3D24Hz%5Ccdot1%7B%2C%7D1m%3D26%7B%2C%7D4%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=f\lambda=24Hz\cdot1{,}1m=26{,}4\ \frac{m}{s}"/></div> <br/> 12-5 <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%3D%5Cfrac%7B24%7D%7B10%5C%20s%7D%3D2%7B%2C%7D4Hz" alt="f=\frac{24}{10\ s}=2{,}4Hz"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=l%3D5%7B%2C%7D7m" alt="l=5{,}7m"/> <div>a) Kumiletkun toinen pää on kiinni seinässä ja toinen pää A jää paikalleen, joten molemmissa päissä on solmukohta. Lisäksi seinän ja pisteen A välissä on kaksi solmua. Solmukohtia on siis yhteensä 4 ja kupuja tulee 3.</div> <div>Piirretään kuva tilanteeseta</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=l%3D3%5Ccdot%5Cfrac%7B%5Clambda%7D%7B2%7D%3D%5Cfrac%7B3%7D%7B2%7D%5Clambda" alt="l=3\cdot\frac{\lambda}{2}=\frac{3}{2}\lambda"/>. Ratkaistaan tästä aallonpistuus</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clambda%3D%5Cfrac%7B2%7D%7B3%7D%5Ccdot%20l%3D%5Cfrac%7B2%7D%7B3%7D%5Ccdot5%7B%2C%7D7m%5Capprox3%7B%2C%7D8m" alt="\lambda=\frac{2}{3}\cdot l=\frac{2}{3}\cdot5{,}7m\approx3{,}8m"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%3D2%7B%2C%7D4Hz%5Ccdot3%7B%2C%7D8m%5Capprox9%7B%2C%7D1%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=f\lambda=2{,}4Hz\cdot3{,}8m\approx9{,}1\ \frac{m}{s}"/></div> <br/> 12-6</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%3D52Hz" alt="f=52Hz"/></div> <div>a)</div> <div>2,5 aallonpituutta</div> <div>b)</div> <div>0,47m</div> <div>c) </div> <div>4</div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%3D52Hz%5Ccdot0%7B%2C%7D47m%3D24%7B%2C%7D44%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=f\lambda=52Hz\cdot0{,}47m=24{,}44\ \frac{m}{s}"/></div> e) <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clambda%3D0%7B%2C%7D47m%5Ccdot2%3D0%7B%2C%7D94m" alt="\lambda=0{,}47m\cdot2=0{,}94m"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D%3D%5Cfrac%7B24%7B%2C%7D44%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B0%7B%2C%7D94%5C%20m%7D%3D26%5C%20Hz" alt="f=\frac{v}{\lambda}=\frac{24{,}44\ \frac{m}{s}}{0{,}94\ m}=26\ Hz"/></div> <br/> <div> <div>12-9 <div> <div> <div>a)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=l%3D%5Cfrac%7B%5Clambda%7D%7B2%7D%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D2l" alt="l=\frac{\lambda}{2}\ \leftrightarrow\ \lambda=2l"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%3Df2l%3D82%7B%2C%7D4%5C%20Hz%5Ccdot2%5Ccdot0%7B%2C%7D65m%3D107%7B%2C%7D12%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=f\lambda=f2l=82{,}4\ Hz\cdot2\cdot0{,}65m=107{,}12\ \frac{m}{s}"/></div> <div>b) </div> <div>Kielen perusvärähtelyssä on kaksi solmua ja yksi kupu, ensimmäisessä ylävärähtelyssä on kolme solmua ja kaksi kupua, joten toisessa on neljä solmua ja kolme kupua. </div> c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=l%3D%5Cfrac%7B5%7D%7B2%7D%5Clambda" alt="l=\frac{5}{2}\lambda"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clambda%3D%5Cfrac%7Bl%7D%7B%5Cfrac%7B5%7D%7B2%7D%7D%3D%5Cfrac%7B65cm%7D%7B2%7B%2C%7D5%7D%3D26%5C%20cm" alt="\lambda=\frac{l}{\frac{5}{2}}=\frac{65cm}{2{,}5}=26\ cm"/></div> </div> <div> <div>12-11</div> <div>a) Koska kiinnityskohdissa lanka ei pääse vapaasti värähtelemään, kiinnityskohtiin muodostuu solmut. </div> <div>b) Perustaajuus <img src="https://math-demo.abitti.fi/math.svg?latex=f%3Df_0" alt="f=f_0"/> vastaa yksinkertaisinta mahdollista seisovan aallon tilannetta. Silloin lankaan muodostuu yksi kupu ja molempiin päihin solmut. Langan pituus on aallonpituuden puolikas: l = λ/2, josta aallonpituudeksi saadaan λ = 2l = 2 ∙ 0,50 m = 1,0 m<br/> c) Värähtelytaajuus on perustaajuuteen verrattuna 4-kertainen eli f = 4 ∙ f0. Lankaan muodostuu 4 aallonpituuden puolikasta ja siksi 4 kupua ja 5 solmua. Langan pituus on <img src="https://math-demo.abitti.fi/math.svg?latex=l%3D4%5Ccdot%5Cfrac%7B%5Clambda%7D%7B2%7D%3D2%5Clambda" alt="l=4\cdot\frac{\lambda}{2}=2\lambda"/> , jostaaallonpituudeksi saadaan <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clambda%3D%5Cfrac%7Bl%7D%7B2%7D%3D%5Cfrac%7B50cm%7D%7B2%7D%3D25cm" alt="\lambda=\frac{l}{2}=\frac{50cm}{2}=25cm"/></div> </div> </div> </div> 2019-05-16T12:41:04+03:00 https://peda.net/id/d90f307e758 2019-05-15T08:40:10+03:00 <span>11-1</span> <div>A 3</div> <div> </div> <div> <div>11-2</div> <div>a,c,d,f</div> <div> </div> <div>11-4</div> <div>Diffraktion takia vesiaalto on tapunut raon kohdalla, ja se taippuu suoraaallosta ympyrämuotoiseksi aalloksi.</div> <div> </div> <div>11-8<br/> 1,0s</div> <div><a href="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-8-1s-png2#top" title="11-8 1s.PNG"><img src="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-8-1s-png2:file/photo/82a829bffcd9c00978e329477420ab742863d11e/11-8%201s.PNG" alt="" title="11-8 1s.PNG" class="inline" loading="lazy"/></a><br/> 2,0s<br/> <a href="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-8-2s-png2#top" title="11-8 2s.PNG"><img src="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-8-2s-png2:file/photo/0bb1a6e53634d0e0cf65e54e10b7fe9ea2a70541/11-8%202s.PNG" alt="" title="11-8 2s.PNG" class="inline" loading="lazy"/></a><br/> 3,0s<br/> <a href="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-8-3s-png2#top" title="11-8 3s.PNG"><img src="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-8-3s-png2:file/photo/091acaedc4f3f5a677cb6afb7e02af67f88d3a2e/11-8%203s.PNG" alt="" title="11-8 3s.PNG" class="inline" loading="lazy"/></a><br/> 4,0s<br/> <a href="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-8-4s-png2#top" title="11-8 4s.PNG"><img src="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-8-4s-png2:file/photo/8637397540c4493924de3d458d7a1f2c1c279108/11-8%204s.PNG" alt="" title="11-8 4s.PNG" class="inline" loading="lazy"/></a><br/> <br/> </div> <div>11-10</div> </div> <div>a) Kun vesiaalto saappuu satama-altaaseen, aalto kimpaa diffraktion vaikutuksesta mereenpäin diffraktion ja tästä syntyy uusi aalto, ja satama-altaan raoista aallto pääsee siitä etenemään, tästäkin syntyy uusi aalto. Uusi aalto saattaa osua edellistä aaltoa, ja tämä on interferenssi.</div> <div>b) <br/> <a href="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-10-b-png#top" title="11-10 b).PNG"><img src="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-10-b-png:file/photo/83a02f2b9d0bb4ff16561ae8d24a125b1dd5c017/11-10%20b%29.PNG" alt="" title="11-10 b).PNG" class="inline" loading="lazy"/></a></div> <div>c)<br/> <a href="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-10-c-png#top" title="11-10 c.PNG"><img src="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-11/11-10-c-png:file/photo/d8353e5faeee6a0d7b1dcc2db3c03817f4d19dd9/11-10%20c.PNG" alt="" title="11-10 c.PNG" class="inline" loading="lazy"/></a></div> 2019-05-13T18:00:29+03:00 https://peda.net/id/0c4a44ba754 2019-05-13T10:36:09+03:00 <div>10-2</div> <div>a) Kun aalto taittuu, taajuus ei muutu, koska aaltoliikkeen taajuus riippuu vain aallon lähteestä.</div> <div>HUOM. Tajuu ei muutu myöskään aallon heijastuessa aineiden rajapinnasta.</div> <div>b) Koska aalto taittuu normaaliin päin, aallon nopeus pienenee.</div> <div>c) Koska aalto taittu normaaliin päin, aallon aallonpituus pienenee</div> <div>HUOM. c-kohdan voi perusteella myös aaltoliikkeen perusyhtälön avulla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda" alt="v=f\lambda"/></div> <br/> 10-4 <div>a)</div> <div>Taittumislaista <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csin%5Calpha_1%7D%7B%5Csin%5Calpha_2%7D%3D%5Cfrac%7Bv_1%7D%7Bv_2%7D" alt="\frac{\sin\alpha_1}{\sin\alpha_2}=\frac{v_1}{v_2}"/>saadaan</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Calpha_2%3D%5Cfrac%7B%5Csin%5Calpha_1%5Ccdot%20v_2%7D%7Bv_1%7D%3D%5Cfrac%7B%5Csin%5Cleft(58%C2%B0%5Cright)%5Ccdot7100%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B14%5C%20100%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%3D0%7B%2C%7D427031...%5Capprox0%7B%2C%7D427" alt="\sin\alpha_2=\frac{\sin\alpha_1\cdot v_2}{v_1}=\frac{\sin\left(58°\right)\cdot7100\ \frac{m}{s}}{14\ 100\ \frac{m}{s}}=0{,}427031...\approx0{,}427"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5E%7B-1%7D%5Cleft(0%7B%2C%7D427%5Cright)%3D25%7B%2C%7D277%5Capprox25%C2%B0" alt="\sin^{-1}\left(0{,}427\right)=25{,}277\approx25°"/></div> <div>b)</div> <div>Taittumislaista <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csin%5Calpha_1%7D%7B%5Csin%5Calpha_2%7D%3D%5Cfrac%7Bv_1%7D%7Bv_2%7D" alt="\frac{\sin\alpha_1}{\sin\alpha_2}=\frac{v_1}{v_2}"/>saadaan </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bv_1%7D%7Bv_2%7D%3D%5Cfrac%7B10%5C%20100%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B11%5C%20200%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%3D0%7B%2C%7D9017...%5Capprox0%7B%2C%7D902" alt="\frac{v_1}{v_2}=\frac{10\ 100\ \frac{m}{s}}{11\ 200\ \frac{m}{s}}=0{,}9017...\approx0{,}902"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5E%7B-1%7D%5Cleft(0%7B%2C%7D902%5Cright)%3D64%7B%2C%7D422...%5Capprox64%C2%B0" alt="\sin^{-1}\left(0{,}902\right)=64{,}422...\approx64°"/></div> <div><br/> <div>10-5</div> <div>a)</div> <div>Koska taitesuhde on 1,9</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csin15%7D%7B%5Csin%5Calpha_2%7D%3D1%7B%2C%7D9" alt="\frac{\sin15}{\sin\alpha_2}=1{,}9"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Calpha_2%3D%5Cfrac%7B%5Csin15%7D%7B1%7B%2C%7D9%7D%3D0%7B%2C%7D1362...%5Capprox0%7B%2C%7D136" alt="\sin\alpha_2=\frac{\sin15}{1{,}9}=0{,}1362...\approx0{,}136"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Calpha_2%3D7%7B%2C%7D816...%5Capprox7%7B%2C%7D8%C2%B0" alt="\alpha_2=7{,}816...\approx7{,}8°"/><br/> <div>b)</div> </div> <div>Taittumislaista <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csin%5Calpha_1%7D%7B%5Csin%5Calpha_2%7D%3D%5Cfrac%7Bv_1%7D%7Bv_2%7D" alt="\frac{\sin\alpha_1}{\sin\alpha_2}=\frac{v_1}{v_2}"/>saadaan  </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v_2%3D%5Cfrac%7B%5Csin%5Calpha_2v_1%7D%7B%5Csin%5Calpha_1%7D%3D%5Cfrac%7B%5Csin%5Cleft(7%7B%2C%7D8%5Cright)25%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B%5Csin%5Cleft(15%5Cright)%7D%3D13%7B%2C%7D109...%5Capprox13%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v_2=\frac{\sin\alpha_2v_1}{\sin\alpha_1}=\frac{\sin\left(7{,}8\right)25\ \frac{m}{s}}{\sin\left(15\right)}=13{,}109...\approx13\ \frac{m}{s}"/></div> <div> <div> <div>10-11</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clambda_1%3D3%7B%2C%7D2cm%3D0%7B%2C%7D032m" alt="\lambda_1=3{,}2cm=0{,}032m"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Clambda_2%3D2%7B%2C%7D6cm%3D0%7B%2C%7D026m" alt="\lambda_2=2{,}6cm=0{,}026m"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Calpha_1%3D30%C2%B0" alt="\alpha_1=30°"/><img src="https://math-demo.abitti.fi/math.svg?latex=v_1%3D7%7B%2C%7D1Hz" alt="v_1=7{,}1Hz"/></div> <div>a) </div> <div>Aaltoliikkeen perusyhtälön <img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda" alt="v=f\lambda"/> mukaan vesiaaltojen nopeus ennen estettä on <img src="https://math-demo.abitti.fi/math.svg?latex=v_1%3Df%5Clambda_1%3D7%7B%2C%7D1Hz%5Ccdot0%7B%2C%7D032m%5Capprox0%7B%2C%7D23%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v_1=f\lambda_1=7{,}1Hz\cdot0{,}032m\approx0{,}23\ \frac{m}{s}"/></div> <div>b)</div> Vesiaaltojen taajuuden määrää aaltolähde, joten taajuus on sama ennen estettä ja esteen päällä eli c)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=f%3D7%7B%2C%7D1Hz" alt="f=7{,}1Hz"/><span>. Vesiaaltojen nopeus esteen päällä on</span><img src="https://math-demo.abitti.fi/math.svg?latex=v_2%3Df%5Clambda_2%3D7%7B%2C%7D1Hz%5Ccdot0%7B%2C%7D026m%5Capprox0%7B%2C%7D18%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v_2=f\lambda_2=7{,}1Hz\cdot0{,}026m\approx0{,}18\ \frac{m}{s}"/><br/> <div>c)</div> <div>Aaltoliikkeen taittumislaki </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Csin%5Calpha_1%7D%7B%5Csin%5Calpha_2%7D%3D%5Cfrac%7Bv_1%7D%7Bv_2%7D" alt="\frac{\sin\alpha_1}{\sin\alpha_2}=\frac{v_1}{v_2}"/>saadaan muotoon </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Csin%5Calpha_2%3D%5Cfrac%7Bv_2%7D%7Bv_1%7D%5Ccdot%5Csin%5Calpha_1%3D%5Cfrac%7Bf%5Clambda_2%7D%7Bf%5Clambda_1%7D%5Ccdot%5Csin%5Calpha%3D%5Cfrac%7B0%7B%2C%7D026m%7D%7B0%7B%2C%7D032m%7D%5Ccdot%5Csin60%C2%B0" alt="\sin\alpha_2=\frac{v_2}{v_1}\cdot\sin\alpha_1=\frac{f\lambda_2}{f\lambda_1}\cdot\sin\alpha=\frac{0{,}026m}{0{,}032m}\cdot\sin60°"/>, josta saadaan taitekulmaksi </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Calpha_2%3D45%C2%B0" alt="\alpha_2=45°"/><br/> <div> </div> </div> 2019-05-13T08:29:15+03:00 https://peda.net/id/bee233ea715 2019-05-08T09:44:36+03:00 <span>9-1</span> <div>a) Ei muuttuu</div> <div>b) Kaksinkertaistuu</div> <div>c) Kun aallonpituus puolittuu, nopeuskin puolittuu</div> <div><br/> <div>9-5</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Clambda%3D100-20%3D80cm" alt="\lambda=100-20=80cm"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20f%3D%5Cfrac%7Bv%7D%7B%5Clambda%7D%3D%5Cfrac%7B25%5C%20%5Cfrac%7Bcm%7D%7Bs%7D%7D%7B80cm%7D%3D0%7B%2C%7D3125%5C%20%5Capprox0%7B%2C%7D31Hz" alt="v=f\lambda\ \leftrightarrow\ f=\frac{v}{\lambda}=\frac{25\ \frac{cm}{s}}{80cm}=0{,}3125\ \approx0{,}31Hz"/></div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=A%3D2%7B%2C%7D0cm" alt="A=2{,}0cm"/></div> <div><br/> <div>9-6</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D%5Cfrac%7Bv%7D%7Bf%7D%3D%5Cfrac%7B4%7B%2C%7D2%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B2%7B%2C%7D1%5C%20Hz%7D%3D2%7B%2C%7D0m" alt="v=f\lambda\ \leftrightarrow\ \lambda=\frac{v}{f}=\frac{4{,}2\ \frac{m}{s}}{2{,}1\ Hz}=2{,}0m"/></div> <div> </div> <div>9-8</div> </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D%5Cfrac%7Bv%7D%7Bf%7D%3D%5Cfrac%7B8%7B%2C%7D0%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B4%7B%2C%7D0%5C%20Hz%7D%3D2%7B%2C%7D0m" alt="v=f\lambda\ \leftrightarrow\ \lambda=\frac{v}{f}=\frac{8{,}0\ \frac{m}{s}}{4{,}0\ Hz}=2{,}0m"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3D%5Cfrac%7Bs%7D%7Bt%7D%5C%20%5Cleftrightarrow%5C%20t%3D%5Cfrac%7B2%7B%2C%7D0m%7D%7B8%7B%2C%7D0%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%3D0%7B%2C%7D25s" alt="v=\frac{s}{t}\ \leftrightarrow\ t=\frac{2{,}0m}{8{,}0\ \frac{m}{s}}=0{,}25s"/><br/> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3Df%5Clambda%5C%20%5Cleftrightarrow%5C%20%5Clambda%3D%5Cfrac%7Bv%7D%7Bf%7D%3D%5Cfrac%7B8%7B%2C%7D0%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%7B4%7B%2C%7D0%5C%20Hz%7D%3D2%7B%2C%7D0m" alt="v=f\lambda\ \leftrightarrow\ \lambda=\frac{v}{f}=\frac{8{,}0\ \frac{m}{s}}{4{,}0\ Hz}=2{,}0m"/> </div> <div>c)</div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B0%7B%2C%7D25s%7D%7B2%7D%3D0%7B%2C%7D125%5Capprox0%7B%2C%7D13s" alt="\frac{0{,}25s}{2}=0{,}125\approx0{,}13s"/> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3D%5Cfrac%7Bs%7D%7Bt%7D%5C%20%5Cleftrightarrow%5C%20t%3D%5Cfrac%7Bs%7D%7Bv%7D%3D%5Cfrac%7B15%7B%2C%7D0m%7D%7B8%7B%2C%7D0%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%3D1%7B%2C%7D875%5Capprox1%7B%2C%7D9s" alt="v=\frac{s}{t}\ \leftrightarrow\ t=\frac{s}{v}=\frac{15{,}0m}{8{,}0\ \frac{m}{s}}=1{,}875\approx1{,}9s"/><br/> <div> </div> </div> 2019-05-08T09:44:36+03:00 https://peda.net/id/c157ca6a6fd 2019-05-08T08:35:32+03:00 <span>8-4</span> <div>a)</div> <div>880 kertaa</div> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B32%7B%2C%7D4%7D%7B23%7D%5Capprox1%7B%2C%7D4s" alt="\frac{32{,}4}{23}\approx1{,}4s"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=f%3D%5Cfrac%7B1%7D%7BT%7D%3D0%7B%2C%7D714...%5Capprox0%7B%2C%7D71Hz" alt="f=\frac{1}{T}=0{,}714...\approx0{,}71Hz"/></div> <div><br/> <div>8-5</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=T%3D2%5Cpi%5Csqrt%5B%5D%7B%5Cfrac%7Bm%7D%7Bk%7D%7D%3D2%5Cpi%5Csqrt%5B%5D%7B%5Cfrac%7B0%7B%2C%7D17kg%7D%7B16%5C%20%5Cfrac%7BN%7D%7Bm%7D%7D%7D%3D0%7B%2C%7D6476...%5Capprox0%7B%2C%7D65s" alt="T=2\pi\sqrt[]{\frac{m}{k}}=2\pi\sqrt[]{\frac{0{,}17kg}{16\ \frac{N}{m}}}=0{,}6476...\approx0{,}65s"/></div> <br/> <div>8-6</div> <div>Koska kappale on kiinnitetty jouseen, siihen kohdistuva voima F on suunta ylös ja painovoima G alas.</div> <div>Newtonin II lain nojallla saadaan yhtälön<img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3D%5Coverline%7B0%7D" alt="\Sigma\overline{F}=\overline{0}"/>eli<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BF%7D%2B%5Coverline%7BG%7D%3D%5Coverline%7B0%7D" alt="\overline{F}+\overline{G}=\overline{0}"/>. Koska <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BF%7D%3D-k%5Coverline%7Bx%7D" alt="\overline{F}=-k\overline{x}"/>, <img src="https://math-demo.abitti.fi/math.svg?latex=-k%5Coverline%7Bx%7D%2B%5Coverline%7BG%7D%3D%5Coverline%7B0%7D" alt="-k\overline{x}+\overline{G}=\overline{0}"/></div> <div>Valitaan suunta alas positiiviseksi, koska sekä <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Bx%7D" alt="\overline{x}"/>että <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BG%7D" alt="\overline{G}"/>ovat positiivisia</div> <div>Yhtälö saadaan, muotoon</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=-kx%2Bmg%3D0" alt="-kx+mg=0"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=-kx%3D-mg" alt="-kx=-mg"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7Bmg%7D%7Bx%7D%3D%5Cfrac%7B0%7B%2C%7D350kg%5Ccdot9%7B%2C%7D81%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%7B0%7B%2C%7D14m%7D%3D24%7B%2C%7D525%5C%20%5Cfrac%7BN%7D%7Bm%7D" alt="k=\frac{mg}{x}=\frac{0{,}350kg\cdot9{,}81\ \frac{m}{s^2}}{0{,}14m}=24{,}525\ \frac{N}{m}"/></div> <div>Värähtelyn jaksonaika saadaan kaavalla</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=T%3D2%5Cpi%5Csqrt%5B%5D%7B%5Cfrac%7B0%7B%2C%7D350kg%7D%7B24%7B%2C%7D525%5C%20%5Cfrac%7BN%7D%7Bm%7D%7D%7D%3D0%7B%2C%7D7506...%5Capprox0%7B%2C%7D75s" alt="T=2\pi\sqrt[]{\frac{0{,}350kg}{24{,}525\ \frac{N}{m}}}=0{,}7506...\approx0{,}75s"/> </div> <div><br/> <div>8-7</div> <div>a)</div> <div>B</div> <div>b) </div> <div>A</div> <div>c)</div> <div>C</div> <div> </div> <div>8-8</div> <div>a)</div> <div>0,91Hz</div> <div>b)</div> </div> <div>Otetaan testipisteeksi pisteet (0,3;0,03) ja (0,2;-0,04)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7By_2-y_1%7D%7Bx_2-x_1%7D%3D%5Cfrac%7B-0%7B%2C%7D04-0%7B%2C%7D03%7D%7B0%7B%2C%7D2-0%7B%2C%7D3%7D%3D0%7B%2C%7D7" alt="k=\frac{y_2-y_1}{x_2-x_1}=\frac{-0{,}04-0{,}03}{0{,}2-0{,}3}=0{,}7"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox0%7B%2C%7D7%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="\approx0{,}7\ \frac{m}{s}"/><br/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=T%3D%5Cfrac%7B1%7D%7Bf%7D%3D%5Cfrac%7B1%7D%7B0%7B%2C%7D91%7Ds" alt="T=\frac{1}{f}=\frac{1}{0{,}91}s"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=T%3D2%5Cpi%5Csqrt%5B%5D%7B%5Cfrac%7Bm%7D%7Bk%7D%7D" alt="T=2\pi\sqrt[]{\frac{m}{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7BT%7D%7B2%5Cpi%7D%3D%5Csqrt%5B%5D%7B%5Cfrac%7Bm%7D%7Bk%7D%7D" alt="\frac{T}{2\pi}=\sqrt[]{\frac{m}{k}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(%5Cfrac%7BT%7D%7B2%5Cpi%7D%5Cright)%5E2%3D%5Cfrac%7Bm%7D%7Bk%7D" alt="\left(\frac{T}{2\pi}\right)^2=\frac{m}{k}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7Bm%7D%7B%5Cleft(%5Cfrac%7BT%7D%7B2%5Cpi%7D%5Cright)%5E2%7D%3D%5Cfrac%7B0%7B%2C%7D500kg%7D%7B%5Cleft(%5Cfrac%7B1%7B%2C%7D0989s%7D%7B2%5Cpi%7D%5Cright)%5E2%7D%3D16%7B%2C%7D346...%5Capprox16%7B%2C%7D35%5C%20%5Cfrac%7BN%7D%7Bm%7D" alt="k=\frac{m}{\left(\frac{T}{2\pi}\right)^2}=\frac{0{,}500kg}{\left(\frac{1{,}0989s}{2\pi}\right)^2}=16{,}346...\approx16{,}35\ \frac{N}{m}"/></div> <div>d) Kuvaajan kärkikohdassa.</div> <div><br/> <div>8-10</div> </div> <div>a)</div> <div>0,67s</div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2A%5Capprox0%7B%2C%7D0544m" alt="2A\approx0{,}0544m"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=A%3D%5Cfrac%7B0%7B%2C%7D0544m%7D%7B2%7D%3D0%7B%2C%7D0272m%5Capprox27%7B%2C%7D2cm" alt="A=\frac{0{,}0544m}{2}=0{,}0272m\approx27{,}2cm"/><br/> <div>c)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D2404...%5Capprox0%7B%2C%7D24%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="0{,}2404...\approx0{,}24\ \frac{m}{s}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D08%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="0{,}08\ \frac{m}{s}"/></div> <div>d)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox1%7B%2C%7D94...%5Capprox2%7B%2C%7D0%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="\approx1{,}94...\approx2{,}0\ \frac{m}{s^2}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Capprox-0%7B%2C%7D369...%5Capprox-0%7B%2C%7D37%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="\approx-0{,}369...\approx-0{,}37\ \frac{m}{s^2}"/> 2019-05-06T11:34:25+03:00 https://peda.net/id/a80f967c6fd 2019-05-06T10:29:45+03:00 <div> <div><span>7-3</span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7BF%7D%7Bm%7D%3D%5Cfrac%7B5%7B%2C%7D0%5C%20N%7D%7B0%7B%2C%7D02%5C%20m%7D%3D250%5C%20%5Cfrac%7BN%7D%7Bm%7D" alt="k=\frac{F}{m}=\frac{5{,}0\ N}{0{,}02\ m}=250\ \frac{N}{m}"/></div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7BF%7D%7Bm%7D%5C%20%5Cleftrightarrow%5C%20m%3D%5Cfrac%7BF%7D%7Bk%7D%3D%5Cfrac%7B15%7B%2C%7D0%5C%20N%7D%7B250%5C%20%5Cfrac%7BN%7D%7Bm%7D%7D%3D0%7B%2C%7D06%5C%20m%3D6%7B%2C%7D0%5C%20cm" alt="k=\frac{F}{m}\ \leftrightarrow\ m=\frac{F}{k}=\frac{15{,}0\ N}{250\ \frac{N}{m}}=0{,}06\ m=6{,}0\ cm"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7BF%7D%7Bm%7D%5C%20%5Cleftrightarrow%5C%20m%3D%5Cfrac%7BF%7D%7Bk%7D%3D%5Cfrac%7B22%7B%2C%7D0%5C%20N%7D%7B250%5C%20%5Cfrac%7BN%7D%7Bm%7D%7D%3D0%7B%2C%7D088%5C%20m%3D8%7B%2C%7D8%5C%20cm" alt="k=\frac{F}{m}\ \leftrightarrow\ m=\frac{F}{k}=\frac{22{,}0\ N}{250\ \frac{N}{m}}=0{,}088\ m=8{,}8\ cm"/></div> <br/> 7-5</div> <div>a)</div> <div>Löysempi</div> <div>b)</div> <div>Jäykämpi</div> <div>c)</div> <div>Jousivoima <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BF%7D%3D-k%5Coverline%7Bx%7D" alt="\overline{F}=-k\overline{x}"/></div> <div>Jousivoima ja poikkeama ovat vastakkaisuuntaisia, joten ottamalla suunnat huomioon ,saadaan <img src="https://math-demo.abitti.fi/math.svg?latex=-F%3D-kx" alt="-F=-kx"/> ja tästä saadaan jousivakioksi <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D%5Cfrac%7BF%7D%7Bx%7D" alt="k=\frac{F}{x}"/></div> <div>Kun jousi katkaistaan keskeltä, on venymä x/2. Tällöin <img src="https://math-demo.abitti.fi/math.svg?latex=k_1%3D%5Cfrac%7BF%7D%7B%5Cfrac%7Bx%7D%7B2%7D%7D%5C%20%5Cleftrightarrow%5C%202k%3D%5Cfrac%7BF%7D%7Bx%7D" alt="k_1=\frac{F}{\frac{x}{2}}\ \leftrightarrow\ 2k=\frac{F}{x}"/></div> <div> </div> <div>7-6</div> <div>a)</div> <div>Jousta puristava voima on -1,7N. Jousivoima <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BF%7D" alt="\overline{F}"/> ja puristava voima ovat yhtä suuret.Mutta vastakkaissuuntaiset, joten jousivoima on <img src="https://math-demo.abitti.fi/math.svg?latex=F%3D1%7B%2C%7D7N" alt="F=1{,}7N"/></div> <div>Koska jousivoima <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BF%7D%3D-k%5Coverline%7Bx%7D" alt="\overline{F}=-k\overline{x}"/>, niin jousivakio on </div> <img src="https://math-demo.abitti.fi/math.svg?latex=k%3D-%5Cfrac%7B%5Coverline%7BF%7D%7D%7Bx%7D%3D-%5Cfrac%7B1%7B%2C%7D7N%7D%7B-0%7B%2C%7D15m%7D%5Capprox11%7B%2C%7D3333%5C%20%5Cfrac%7BN%7D%7Bm%7D" alt="k=-\frac{\overline{F}}{x}=-\frac{1{,}7N}{-0{,}15m}\approx11{,}3333\ \frac{N}{m}"/></div> <div>Newtonin II-lain mukaan <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D" alt="\Sigma\overline{F}=m\overline{a}"/></div> <div>Kun vaunu päästetään irti, vaikuttaa siihen tason suunnassa ainoastaan jousvoima, koska kitkaa ja ilmanvastusta ei oteta huomioon. (Pystysuunnassa vaunulla ei ole kiihtyvyyttä)</div> <div>Näin ollen <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D" alt="\overline{F}=m\overline{a}"/>. Kiihtyvyys on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D%3D%5Cfrac%7B%5Coverline%7BF%7D%7D%7Bm%7D%3D%5Cfrac%7B-k%5Coverline%7Bx%7D%7D%7Bm%7D" alt="\overline{a}=\frac{\overline{F}}{m}=\frac{-k\overline{x}}{m}"/></div> <div>a) Tasapainoasemassa x=0,00m, jousivoima on nolla, joten kiihtyvyys on <img src="https://math-demo.abitti.fi/math.svg?latex=a%3D0%7B%2C%7D0%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="a=0{,}0\ \frac{m}{s^2}"/></div> <div>b) Hetki ennen vaunun irtipäästöä jousta puristava voima ja jousivoima ovat yhtä suuret ja vastakkaisuuntaiset. Vaunnu lähtee liikkeelle ääriasennosta ja sen kiihtyvyys ja jousivoima ovat samansuuntaiset. Vaunun kiihtyvyys on<img src="https://math-demo.abitti.fi/math.svg?latex=a%3D%5Cfrac%7BF%7D%7Bm%7D%3D%5Cfrac%7B1%7B%2C%7D7%5C%20N%7D%7B0%7B%2C%7D76%5C%20kg%7D%5Capprox2%7B%2C%7D2%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="a=\frac{F}{m}=\frac{1{,}7\ N}{0{,}76\ kg}\approx2{,}2\ \frac{m}{s^2}"/></div> <div>c) x=0,12m </div> <div>Vaunu on ohttanut tasapainoaseman. Jousivoima suuntautuu kohti tasapainoasemaa. Vaunun kiihtyvyys on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a%3D%5Cfrac%7B-k%5Coverline%7Bx%7D%7D%7Bm%7D%3D%5Cfrac%7B-11%7B%2C%7D3333%5C%20%5Cfrac%7BN%7D%7Bm%7D%5Ccdot0%7B%2C%7D12m%7D%7B0%7B%2C%7D76kg%7D%5Capprox-1%7B%2C%7D8%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="a=\frac{-k\overline{x}}{m}=\frac{-11{,}3333\ \frac{N}{m}\cdot0{,}12m}{0{,}76kg}\approx-1{,}8\ \frac{m}{s^2}"/></div> <div>Vaunu on kiihtyvässä liikkeessä kohti tasapainoasemaa tai hidastuvassa liikeessä alkuperäoseen positiiviseen liikesuuntaan nähden.</div> 2019-05-06T10:29:17+03:00 https://peda.net/id/3d237eb66a5 2019-05-03T14:31:34+03:00 <div>6-1</div> <div>a) </div> <div>Lasketaan avaruusaseman ratanopeus</div> <div>Gravitaatiovuorovaikutus pitää sateliitin radallaan. Newtonin II lain mukaan on<span> </span><img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D_n" alt="\Sigma\overline{F}=m\overline{a}_n"/>, jossa gravitaatiovoiman suuruus on <img src="https://math-demo.abitti.fi/math.svg?latex=F%3D%5Cgamma%5C%20%5Cfrac%7Bm_1m_2%7D%7Br%5E2%7D" alt="F=\gamma\ \frac{m_1m_2}{r^2}"/>ja normaalikiihtyvyyden suuruus <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="a_n=\frac{v^2}{r}"/>. Yhtälöstä <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5C%20%5Cfrac%7Bm_1m_2%7D%7Br%5E2%7D%3D%5Cfrac%7Bm_2v%5E2%7D%7Br%7D" alt="\gamma\ \frac{m_1m_2}{r^2}=\frac{m_2v^2}{r}"/>ratanopeudeksi saadaan </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5C%20%5Cfrac%7BmM%7D%7Br%5E2%7D%3D%5Cfrac%7Bmv%5E2%7D%7Br%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Ccdot%5Cfrac%7Br%7D%7Bm%7D" alt="\gamma\ \frac{mM}{r^2}=\frac{mv^2}{r}\ \ \ \ \ \left|\right|\cdot\frac{r}{m}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=v%5E2%3D%5Cfrac%7B%5Cgamma%20M%7D%7Br%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Csqrt%5B%5D%7B%7D" alt="v^2=\frac{\gamma M}{r}\ \ \ \ \ \left|\right|\sqrt[]{}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3D%5Csqrt%5B%5D%7B%5Cfrac%7B%5Cgamma%20M%7D%7Br%7D%7D%3D%5Csqrt%5B%5D%7B%5Cfrac%7B6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%7D%7B410%5C%20000%5C%20m%2B6%5C%20378%5C%20140%5C%20m%5C%20%7D%7D%3D7664%7B%2C%7D0693...%5Capprox7664%7B%2C%7D07%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=\sqrt[]{\frac{\gamma M}{r}}=\sqrt[]{\frac{6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}\cdot5{,}974\cdot10^{24}kg}{410\ 000\ m+6\ 378\ 140\ m\ }}=7664{,}0693...\approx7664{,}07\ \frac{m}{s}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=E_k%3D%5Cfrac%7B1%7D%7B2%7Dmv%5E2%3D%5Cfrac%7B1%7D%7B2%7D%5Ccdot420%5C%20000kg%5Ccdot%5Cleft(7664%7B%2C%7D07%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Cright)%5E2%3D1%7B%2C%7D233...%5Ccdot10%5E%7B13%7D%5Capprox1%7B%2C%7D2%5Ccdot10%5E%7B13%7DJ" alt="E_k=\frac{1}{2}mv^2=\frac{1}{2}\cdot420\ 000kg\cdot\left(7664{,}07\ \frac{m}{s}\right)^2=1{,}233...\cdot10^{13}\approx1{,}2\cdot10^{13}J"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=E_p%3D-%5Cgamma%5Cfrac%7BmM%7D%7Br%7D%3D-6%7B%2C%7D67429%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot%5Cfrac%7B420%5C%20000kg%5Ccdot5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%7D%7B6%5C%20378%5C%20140%5C%20m%5C%20%7D" alt="E_p=-\gamma\frac{mM}{r}=-6{,}67429\cdot10^{-11}\ \frac{Nm^2}{kg^2}\cdot\frac{420\ 000kg\cdot5{,}974\cdot10^{24}kg}{6\ 378\ 140\ m\ }"/></div> <div>c)</div> <div><br/> <div>6-2 </div> <div>m=satelliitin massa, M=Maan massa</div> <div>r=Sateliitin etäisyys Maan keskipisteestä</div> <div>Sateliitin liike-energia on <img src="https://math-demo.abitti.fi/math.svg?latex=E_k%3D%5Cfrac%7B1%7D%7B2%7Dmv%5E2" alt="E_k=\frac{1}{2}mv^2"/>, missä v=satelliitin ratanopeus</div> <div>Newtonin II-lain mukaan </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D_n" alt="\Sigma\overline{F}=m\overline{a}_n"/></div> <div>Gravitaatiovuorovaikutus pitää satelliitin radallaan</div> <div>Gravitaatiovoima on <img src="https://math-demo.abitti.fi/math.svg?latex=F%3D%5Cgamma%5Cfrac%7BmM%7D%7Br%5E2%7D" alt="F=\gamma\frac{mM}{r^2}"/> ja normaalikiihtyvyys on <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="a_n=\frac{v^2}{r}"/></div> <div>Ratkaistaan sateliitin ratanopeuden neliö yhtälöstä</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5C%20%5Cfrac%7BmM%7D%7Br%5E2%7D%3D%5Cfrac%7Bmv%5E2%7D%7Br%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Ccdot%5Cfrac%7Br%7D%7Bm%7D" alt="\gamma\ \frac{mM}{r^2}=\frac{mv^2}{r}\ \ \ \ \ \left|\right|\cdot\frac{r}{m}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cgamma%20M%7D%7Br%7D%3Dv%5E2" alt="\frac{\gamma M}{r}=v^2"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=E_k%3D%5Cfrac%7B1%7D%7B2%7Dmv%5E2%3D%5Cfrac%7B1%7D%7B2%7Dm%5Ccdot%5Cfrac%7B%5Cgamma%20M%7D%7Br%7D%3D%5Cfrac%7B%5Cgamma%20mM%7D%7B2r%7D" alt="E_k=\frac{1}{2}mv^2=\frac{1}{2}m\cdot\frac{\gamma M}{r}=\frac{\gamma mM}{2r}"/><br/> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%5BE_k%5Cright%5D%3D%5Cfrac%7B%5Cleft%5B%5Cgamma%5Cright%5D%5Cleft%5Bm%5Cright%5D%5Cleft%5BM%5Cright%5D%7D%7B%5Cleft%5B2%5Cright%5D%5Cleft%5Br%5Cright%5D%7D%3D%5Cfrac%7B1%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot1kg%5Ccdot1kg%7D%7B1m%7D%3D1%5Ccdot%5Cfrac%7BNm%5E2%7D%7Bm%7D%3D1Nm%3D1J" alt="\left[E_k\right]=\frac{\left[\gamma\right]\left[m\right]\left[M\right]}{\left[2\right]\left[r\right]}=\frac{1\ \frac{Nm^2}{kg^2}\cdot1kg\cdot1kg}{1m}=1\cdot\frac{Nm^2}{m}=1Nm=1J"/></div> <br/> 6-4 <div><span>a)</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=W%3DE_%7Bp%7B%2C%7Dl%7D-E_%7Bp%7B%2C%7Da%7D" alt="W=E_{p{,}l}-E_{p{,}a}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=W%3D%5Cleft(-%5Cgamma%5Ccdot%5Cfrac%7BmM%7D%7Br_2%7D%5Cright)-%5Cleft(-%5Cgamma%5Ccdot%5Cfrac%7BmM%7D%7Br_1%7D%5Cright)%3D-%5Cgamma%5Ccdot%5Cfrac%7BmM%7D%7Br_2%7D%2B%5Cgamma%5Ccdot%5Cfrac%7BmM%7D%7Br_1%7D%3D-%5Cgamma%20mM%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7Br_2%7D-%5Cfrac%7B1%7D%7Br_1%7D%5Cright)" alt="W=\left(-\gamma\cdot\frac{mM}{r_2}\right)-\left(-\gamma\cdot\frac{mM}{r_1}\right)=-\gamma\cdot\frac{mM}{r_2}+\gamma\cdot\frac{mM}{r_1}=-\gamma mM\cdot\left(\frac{1}{r_2}-\frac{1}{r_1}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=W%3D-6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot3170kg%5Ccdot5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%5Ccdot%5Cleft(%5Cfrac%7B1%7D%7B410%5C%20km%2B6378%7B%2C%7D140%5C%20km%7D-%5Cfrac%7B1%7D%7B6378%7B%2C%7D140%5C%20km%7D%5Cright)" alt="W=-6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}\cdot3170kg\cdot5{,}974\cdot10^{24}kg\cdot\left(\frac{1}{410\ km+6378{,}140\ km}-\frac{1}{6378{,}140\ km}\right)"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=W%3D11%7B%2C%7D9698GJ%5Capprox12GJ" alt="W=11{,}9698GJ\approx12GJ"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=E_p%3Dmgh%3D3170kg%5Ccdot9%7B%2C%7D81%5Cfrac%7Bm%7D%7Bs%5E2%7D%5Ccdot410km%5Capprox12%7B%2C%7D7501GJ" alt="E_p=mgh=3170kg\cdot9{,}81\frac{m}{s^2}\cdot410km\approx12{,}7501GJ"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B12%7B%2C%7D7501GJ-11%7B%2C%7D9698GJ%7D%7B11%7B%2C%7D9678GJ%7D%5Capprox6%7B%2C%7D5%5C%25" alt="\frac{12{,}7501GJ-11{,}9698GJ}{11{,}9678GJ}\approx6{,}5\%"/><br/> <br/> 6-6<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=E_%7Bk%7B%2C%7Da%7D%2BE_%7Bp%7B%2C%7Da%7D%3DE_%7Bk%7B%2C%7Dl%7D%2BE_%7Bp%7B%2C%7Dl%7D" alt="E_{k{,}a}+E_{p{,}a}=E_{k{,}l}+E_{p{,}l}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B2%7Dmv_a%5E2%2B%5Cleft(-%5Cgamma%5Cfrac%7BmM%7D%7Br%7D%5Cright)%3D%5Cfrac%7B1%7D%7B2%7Dmv_l%5E2%2B%5Cleft(-%5Cgamma%5Cfrac%7BmM%7D%7Brl%7D%5Cright)" alt="\frac{1}{2}mv_a^2+\left(-\gamma\frac{mM}{r}\right)=\frac{1}{2}mv_l^2+\left(-\gamma\frac{mM}{rl}\right)"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B2%7Dmv_a%5E2%2B%5Cleft(-%5Cgamma%5Cfrac%7BmM%7D%7BR%7D%5Cright)%3D%5C%20-%5Cgamma%5Cfrac%7BmM%7D%7Br%7D" alt="\frac{1}{2}mv_a^2+\left(-\gamma\frac{mM}{R}\right)=\ -\gamma\frac{mM}{r}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cgamma%20M%7D%7Br%7D%3D%5Cfrac%7B%5Cgamma%20M%7D%7BR%7D-%5Cfrac%7B1%7D%7B2%7Dv_a%5E2" alt="\frac{\gamma M}{r}=\frac{\gamma M}{R}-\frac{1}{2}v_a^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=r%3D%5Cfrac%7B%5Cgamma%20M%7D%7B%5Cfrac%7B%5Cgamma%20M%7D%7BR%7D-%5Cfrac%7B1%7D%7B2%7Dv_a%5E2%7D%3D%5Cfrac%7B6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNM%5E2%7D%7Bkg%5E2%7D%5Ccdot5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%7D%7B%5Cfrac%7B6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNM%5E2%7D%7Bkg%5E2%7D%5Ccdot5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%7D%7B6378%5Ccdot10%5E3m%7D-%5Cfrac%7B1%7D%7B2%7D%5Ccdot%5Cleft(8%7B%2C%7D5%5Ccdot10%5E3%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Cright)%5E2%3D15%5C%20108%5C%20736m%5Capprox15%5C%20108%7B%2C%7D7%5C%20km%7D" alt="r=\frac{\gamma M}{\frac{\gamma M}{R}-\frac{1}{2}v_a^2}=\frac{6{,}67428\cdot10^{-11}\ \frac{NM^2}{kg^2}\cdot5{,}974\cdot10^{24}kg}{\frac{6{,}67428\cdot10^{-11}\ \frac{NM^2}{kg^2}\cdot5{,}974\cdot10^{24}kg}{6378\cdot10^3m}-\frac{1}{2}\cdot\left(8{,}5\cdot10^3\ \frac{m}{s}\right)^2=15\ 108\ 736m\approx15\ 108{,}7\ km}"/></div> <div>6-8<br/> Kaikki oikein</div> </div> 2019-04-29T11:46:11+03:00 https://peda.net/id/1e9f0d52674 2019-04-29T19:39:34+03:00 <span><span>5-2<br/> </span></span> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="a_n=\frac{v^2}{r}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3D29%7B%2C%7D78%5C%20%5Cfrac%7Bkm%7D%7Bs%7D%3D29780%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=29{,}78\ \frac{km}{s}=29780\ \frac{m}{s}"/></div> <span><span><img src="https://math-demo.abitti.fi/math.svg?latex=r%3D149%7B%2C%7D59787%5Ccdot10%5E9m" alt="r=149{,}59787\cdot10^9m"/><br/> </span></span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B%5Cleft(29780%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Cright)%5E2%7D%7B149%7B%2C%7D59787%5Ccdot10%5E6km%7D%3D5%7B%2C%7D928...%5Ccdot10%5E%7B-3%7D%5Capprox5%7B%2C%7D93%5Ccdot10%5E%7B-3%7D%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="a_n=\frac{\left(29780\ \frac{m}{s}\right)^2}{149{,}59787\cdot10^6km}=5{,}928...\cdot10^{-3}\approx5{,}93\cdot10^{-3}\ \frac{m}{s^2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="a_n=\frac{v^2}{r}"/> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r%3D384%5C%20400km%3D384%5C%20400%5C%20000m" alt="r=384\ 400km=384\ 400\ 000m"/></div> <span><span><img src="https://math-demo.abitti.fi/math.svg?latex=v%3D1%7B%2C%7D023%5Cfrac%7Bkm%7D%7Bs%7D%3D1023%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=1{,}023\frac{km}{s}=1023\ \frac{m}{s}"/></span></span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7B%5Cleft(1023%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Cright)%5E2%7D%7B384%5C%20400%5C%20000m%7D%3D2%7B%2C%7D7225%5Ccdot10%5E%7B-3%7D%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="a_n=\frac{\left(1023\ \frac{m}{s}\right)^2}{384\ 400\ 000m}=2{,}7225\cdot10^{-3}\ \frac{m}{s^2}"/></div> <span><br/> 5-4</span> <div>Maan massa on <img src="https://math-demo.abitti.fi/math.svg?latex=M%3D5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg" alt="M=5{,}974\cdot10^{24}kg"/></div> <div>Kuun massa on <img src="https://math-demo.abitti.fi/math.svg?latex=M%3D7%7B%2C%7D348%5Ccdot10%5E%7B22%7Dkg" alt="M=7{,}348\cdot10^{22}kg"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%3D6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D" alt="\gamma=6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}"/></div> <div>Maan ja kuun keskietäisyys </div> <img src="https://math-demo.abitti.fi/math.svg?latex=r%3D384%5C%20400%5Ccdot10%5E3m" alt="r=384\ 400\cdot10^3m"/><br/> <div>a) Gravitaatiovuorovaikutus pitää Kuun Maata kiertävällä radalla, joka voidaan mallintaa ympyräratana</div> <div>Newtonin II-lain mukaan</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D_n" alt="\Sigma\overline{F}=m\overline{a}_n"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%3D%5Cgamma%5Cfrac%7BMm%7D%7Br%5E2%7D" alt="F=\gamma\frac{Mm}{r^2}"/> <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="a_n=\frac{v^2}{r}"/></div> <div>Sijoittaalla saadaan</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5Cfrac%7BMm%7D%7Br%5E2%7D%3Dm%5Cfrac%7Bv%5E2%7D%7Br%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3Am" alt="\gamma\frac{Mm}{r^2}=m\frac{v^2}{r}\ \ \ \ \ \left|\right|:m"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5Ccdot%5Cfrac%7BM%7D%7Br%5E2%7D%3D%5Cfrac%7Bv%5E2%7D%7Br%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Ccdot%20r" alt="\gamma\cdot\frac{M}{r^2}=\frac{v^2}{r}\ \ \ \ \ \left|\right|\cdot r"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=v%5E2%3D%5Cfrac%7B%5Cgamma%20M%7D%7Br%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Csqrt%5B%5D%7B%7D" alt="v^2=\frac{\gamma M}{r}\ \ \ \ \ \left|\right|\sqrt[]{}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3D%5Csqrt%5B%5D%7B%5Cfrac%7B%5Cgamma%20M%7D%7Br%7D%7D%3D1018%7B%2C%7D46%5C%20%5Cfrac%7Bm%7D%7Bs%7D%3D1%7B%2C%7D018%5C%20%5Cfrac%7Bkm%7D%7Bs%7D" alt="v=\sqrt[]{\frac{\gamma M}{r}}=1018{,}46\ \frac{m}{s}=1{,}018\ \frac{km}{s}"/><br/> <div>b) Koska Kuu on ympyräradalla, yksi kierros on pituudeltaan 2πr. </div> <div>Siten kierrtoaika on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=t%3D%5Cfrac%7Bs%7D%7Bv%7D%3D%5Cfrac%7B2%5Cpi%5Ccdot384400%5Ccdot10%5E3m%7D%7B1019%7B%2C%7D46%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%3D2371479s" alt="t=\frac{s}{v}=\frac{2\pi\cdot384400\cdot10^3m}{1019{,}46\ \frac{m}{s}}=2371479s"/><br/> <br/> 5-5<br/> <div> <div>a)</div> <div>Sateliitin kiertoaika planeetan ympäri on 7h39min=27540s</div> <div>Sateliitin kiertoradan pituus on <img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cpi%20r%3D2%5Cpi%5Ccdot9%5C%20370%5C%20000%5C%20m%3D58%7B%2C%7D8734%5Ccdot10%5E6m" alt="2\pi r=2\pi\cdot9\ 370\ 000\ m=58{,}8734\cdot10^6m"/>ja ratanopeus</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3D%5Cfrac%7B2%5Cpi%20r%7D%7BT%7D%3D%5Cfrac%7B58%7B%2C%7D934%5Ccdot10%5E6m%7D%7B27%5C%20540s%7D%5Capprox2137%7B%2C%7D74%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=\frac{2\pi r}{T}=\frac{58{,}934\cdot10^6m}{27\ 540s}\approx2137{,}74\ \frac{m}{s}"/></div> <div>Gravitaatiovuorovaikutus pitää sateliitin radallaan . Newtonin II lain mukaan <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm_2%5Coverline%7Ba%7D" alt="\Sigma\overline{F}=m_2\overline{a}"/>, jossa gravitaatiovoiman suuruus on <img src="https://math-demo.abitti.fi/math.svg?latex=F%3D%5Cgamma%5C%20%5Cfrac%7Bm_1m_2%7D%7Br%5E2%7D" alt="F=\gamma\ \frac{m_1m_2}{r^2}"/>ja normaalikiihtyvyyden suuruus <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="a_n=\frac{v^2}{r}"/>. Yhtälöstä <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5C%20%5Cfrac%7Bm_1m_2%7D%7Br%5E2%7D%3D%5Cfrac%7Bm_2v%5E2%7D%7Br%7D" alt="\gamma\ \frac{m_1m_2}{r^2}=\frac{m_2v^2}{r}"/>planeetan massaksi saadaan</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=m_1%3D%5Cfrac%7Bv%5E2r%7D%7B%5Cgamma%7D%3D%5Cfrac%7B%5Cleft(2137%7B%2C%7D74%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Cright)%5E2%5Ccdot9%5C%20370%5C%20000m%7D%7B6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%7D%5Capprox6%7B%2C%7D42%5Ccdot10%5E%7B23%5C%20kg.%7D" alt="m_1=\frac{v^2r}{\gamma}=\frac{\left(2137{,}74\ \frac{m}{s}\right)^2\cdot9\ 370\ 000m}{6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}}\approx6{,}42\cdot10^{23\ kg.}"/></div> </div> <br/> 5-6 <div> <div>Gravitaatiovuorovaikutus pitää sateliitin radallaan. Newtonin II lain mukaan on <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D_n" alt="\Sigma\overline{F}=m\overline{a}_n"/>, jossa gravitaatiovoiman suuruus on <span> </span><img src="https://math-demo.abitti.fi/math.svg?latex=F%3D%5Cgamma%5C%20%5Cfrac%7Bm_1m_2%7D%7Br%5E2%7D" alt="F=\gamma\ \frac{m_1m_2}{r^2}"/>ja normaalikiihtyvyyden suuruus <img src="https://math-demo.abitti.fi/math.svg?latex=a_n%3D%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="a_n=\frac{v^2}{r}"/>. Yhtälöstä <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5C%20%5Cfrac%7Bm_1m_2%7D%7Br%5E2%7D%3D%5Cfrac%7Bm_2v%5E2%7D%7Br%7D" alt="\gamma\ \frac{m_1m_2}{r^2}=\frac{m_2v^2}{r}"/>ratanopeudeksi saadaan</div> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3D%5Csqrt%5B%5D%7B%5Cfrac%7B%5Cgamma%20M%7D%7Br%7D%7D%3D%5Csqrt%5B%5D%7B%5Cfrac%7B6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%7D%7B6378km%2B610km%7D%7D%3D7553%7B%2C%7D68%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Capprox7%7B%2C%7D6%5C%20%5Cfrac%7Bkm%7D%7Bs%7D" alt="v=\sqrt[]{\frac{\gamma M}{r}}=\sqrt[]{\frac{6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}\cdot5{,}974\cdot10^{24}kg}{6378km+610km}}=7553{,}68\ \frac{m}{s}\approx7{,}6\ \frac{km}{s}"/></div> <div>b) kiertoaika Maan ympäri on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=t%3D%5Cfrac%7Bs%7D%7Bv%7D%3D%5Cfrac%7B2%5Cpi%5Ccdot%5Cleft(6378%5Ccdot10%5E3m%2B610%5Ccdot10%5E3m%5Cright)%7D%7B7553%7B%2C%7D68%5C%20%5Cfrac%7Bm%7D%7Bs%7D%7D%3D5812%7B%2C%7D65s%5Capprox97%5Cmin" alt="t=\frac{s}{v}=\frac{2\pi\cdot\left(6378\cdot10^3m+610\cdot10^3m\right)}{7553{,}68\ \frac{m}{s}}=5812{,}65s\approx97\min"/></div> <div>c) Kulmanopeus on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Comega%3D%5Cfrac%7B%5CDelta%5Cvarphi%7D%7B%5CDelta%20t%7D%3D%5Cfrac%7B2%5Cpi%7D%7B5812%7B%2C%7D65s%7D%5Capprox1%7B%2C%7D1%5Ccdot10%5E%7B-3%7D%5C%20%5Cfrac%7Brad%7D%7Bs%7D" alt="\omega=\frac{\Delta\varphi}{\Delta t}=\frac{2\pi}{5812{,}65s}\approx1{,}1\cdot10^{-3}\ \frac{rad}{s}"/></div> <br/> <span>5-12</span> <div>a)</div> <div>Epätosi</div> <div>b)</div> <div>Tosi</div> <div>c) </div> <div>Tosi</div> <div>d) </div> <div>Tosi</div> </div> 2019-04-25T14:06:05+03:00 https://peda.net/id/cd59e702667 2019-04-25T12:53:31+03:00 <span><span>4-2<br/> Vaikka astronautti tuntee itensä painottomaksi, häneen ja avaruusalukseen kohdistuu man vetovoima. sillä avanruusalus pysyy maata kiertävällä radalla. Eäisyys on kuitenkin sen verrean suuri, että avaruusalus ei putoa maahan ja sen verran pieni, ettei avaruusaluus leijaile ulkoavaruuteen.<br/> <br/> 4-5<br/> </span></span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r_%7Bmaa%7D%3D6380km%3D6%5C%20380%5C%20000m" alt="r_{maa}=6380km=6\ 380\ 000m"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%3D6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D" alt="\gamma=6{,}67428\cdot10^{-11}\frac{Nm^2}{kg^2}"/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=M%3D5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg" alt="M=5{,}974\cdot10^{24}kg"/><span> </span></div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g_%7Bmaa%7D%3D%5Cgamma%5Cfrac%7BM%7D%7Br%5E2%7D%3D6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot%5Cfrac%7B5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%7D%7B%5Cleft(6%5C%20380%5C%20000m%5Cright)%5E2%7D%3D9%7B%2C%7D7955377...%5Capprox9%7B%2C%7D80%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="g_{maa}=\gamma\frac{M}{r^2}=6{,}67428\cdot10^{-11}\frac{Nm^2}{kg^2}\cdot\frac{5{,}974\cdot10^{24}kg}{\left(6\ 380\ 000m\right)^2}=9{,}7955377...\approx9{,}80\ \frac{m}{s^2}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r_%7B%2BMount%5C%20Kenyan%7D%3D6%5C%20380%5C%20000m%2B5199m%3D6385199m" alt="r_{+Mount\ Kenyan}=6\ 380\ 000m+5199m=6385199m"/></div> <span><span><img src="https://math-demo.abitti.fi/math.svg?latex=g_%7Bmaa%7D%3D%5Cgamma%5Cfrac%7BM%7D%7Br%5E2%7D%3D6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot%5Cfrac%7B5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%7D%7B%5Cleft(6385199m%5Cright)%5E2%7D%3D9%7B%2C%7D779...%5Capprox9%7B%2C%7D78%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="g_{maa}=\gamma\frac{M}{r^2}=6{,}67428\cdot10^{-11}\frac{Nm^2}{kg^2}\cdot\frac{5{,}974\cdot10^{24}kg}{\left(6385199m\right)^2}=9{,}779...\approx9{,}78\ \frac{m}{s^2}"/></span></span> <div>c)</div> <div>Pienenee</div> <span><span><br/> 4-6<br/> </span></span> <div>a) </div> <span><span><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7BF%7D%7Bm%7D%3Dg" alt="\frac{F}{m}=g"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%3D9%7B%2C%7D766%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="g=9{,}766\ \frac{m}{s^2}"/></span></span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=m%3D%5Cfrac%7BF%7D%7Bg%7D%3D%5Cfrac%7B539%7B%2C%7D8N%7D%7B9%7B%2C%7D825%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%3D54%7B%2C%7D9414...%5Capprox54%7B%2C%7D94kg" alt="m=\frac{F}{g}=\frac{539{,}8N}{9{,}825\ \frac{m}{s^2}}=54{,}9414...\approx54{,}94kg"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F_%7BMexico%7D%3Dmg%3D54%7B%2C%7D94kg%5Ccdot9%7B%2C%7D766%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%3D536%7B%2C%7D5584529...%5Capprox536%7B%2C%7D56N" alt="F_{Mexico}=mg=54{,}94kg\cdot9{,}766\ \frac{m}{s^2}=536{,}5584529...\approx536{,}56N"/><br/> <div>b)</div> </div> <div>53kg</div> <span><br/> 4-8<br/> </span> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r%3D5%7B%2C%7D7km" alt="r=5{,}7km"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Crho%3D9%7B%2C%7D2%5Ccdot10%5E%7B17%7D%5C%20%5Cfrac%7Bkg%7D%7Bm%5E3%7D" alt="\rho=9{,}2\cdot10^{17}\ \frac{kg}{m^3}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%3D6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D" alt="\gamma=6{,}67428\cdot10^{-11}\frac{Nm^2}{kg^2}"/></div> <div>Gravitaatiokentän voimakkuus neutronitähden pinnalla on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g_%7Bt%C3%A4hti%7D%3D%5Cgamma%5Cfrac%7BM%7D%7Br%5E2%7D" alt="g_{tähti}=\gamma\frac{M}{r^2}"/></div> <div>Tähden massa on <img src="https://math-demo.abitti.fi/math.svg?latex=m%3D%5Crho%20V" alt="m=\rho V"/></div> <div>Oletetaan tähti pallon muotoiseksi, jolloin sen tilavuus on <img src="https://math-demo.abitti.fi/math.svg?latex=V%3D%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3" alt="V=\frac{4}{3}\pi r^3"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=g_%7Bt%C3%A4hti%7D%3D%5Cgamma%5Cfrac%7B%5Crho%5Ccdot%5Cfrac%7B4%7D%7B3%7D%5Ccdot%5Cpi%20r%5E3%7D%7Br%5E2%7D%3D%5Cgamma%5Ccdot%5Crho%5Ccdot%5Cfrac%7B4%7D%7B3%7D%5Ccdot%5Cpi%20r%3D6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot9%7B%2C%7D2%5Ccdot10%5E%7B17%7D%5C%20%5Cfrac%7Bkg%7D%7Bm%5E3%7D%5Ccdot%5Cfrac%7B4%7D%7B3%7D%5Ccdot%5Cpi%5Ccdot5700m%5Capprox1%7B%2C%7D5%5Ccdot10%5E%7B12%7D%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="g_{tähti}=\gamma\frac{\rho\cdot\frac{4}{3}\cdot\pi r^3}{r^2}=\gamma\cdot\rho\cdot\frac{4}{3}\cdot\pi r=6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}\cdot9{,}2\cdot10^{17}\ \frac{kg}{m^3}\cdot\frac{4}{3}\cdot\pi\cdot5700m\approx1{,}5\cdot10^{12}\ \frac{m}{s^2}"/><br/> <div>4-9<br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%3Da" alt="g=a"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%3D9%7B%2C%7D80665%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="g=9{,}80665\ \frac{m}{s^2}"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%3D6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D" alt="\gamma=6{,}67428\cdot10^{-11}\frac{Nm^2}{kg^2}"/> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r_%7Bmaa%7D%3D6380km%3D6%5C%20380%5C%20000m" alt="r_{maa}=6380km=6\ 380\ 000m"/> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=M%3D5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg" alt="M=5{,}974\cdot10^{24}kg"/>  </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1%7D%7B2%7Dg%3D4%7B%2C%7D903325%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="\frac{1}{2}g=4{,}903325\ \frac{m}{s^2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g%3D%5Cgamma%5C%20%5Ccdot%5Cfrac%7BM%7D%7Br%5E2%7D%5C%20%5Cleftrightarrow%5C%20r%3D%5Csqrt%5B%5D%7B%5Cfrac%7B%5Cgamma%20M%7D%7Bg%7D%7D" alt="g=\gamma\ \cdot\frac{M}{r^2}\ \leftrightarrow\ r=\sqrt[]{\frac{\gamma M}{g}}"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r%3D%5Csqrt%5B%5D%7B%5Cfrac%7B%5Cgamma%20M%7D%7Bg%7D%7D%3D%5Csqrt%5B%5D%7B%5Cfrac%7B6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot5%7B%2C%7D974%5Ccdot10%5E%7B24%7Dkg%7D%7B4%7B%2C%7D903325%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%7D%3D9017638%7B%2C%7D095m" alt="r=\sqrt[]{\frac{\gamma M}{g}}=\sqrt[]{\frac{6{,}67428\cdot10^{-11}\frac{Nm^2}{kg^2}\cdot5{,}974\cdot10^{24}kg}{4{,}903325\ \frac{m}{s^2}}}=9017638{,}095m"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=9017638%7B%2C%7D095m-6%5C%20380%5C%20000m%3D2637638%7B%2C%7D095m%5Capprox2637%7B%2C%7D638095km%5Capprox2600km" alt="9017638{,}095m-6\ 380\ 000m=2637638{,}095m\approx2637{,}638095km\approx2600km"/></div> </div> <br/> <span>4-12</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=g_r%3D%5Cgamma%5Cfrac%7BM%7D%7Br%5E2%7D%5C%20%5Cleftrightarrow%20r%3D%5Csqrt%5B%5D%7B%5Cfrac%7B%5Cgamma%20M%7D%7Bg_r%7D%7D" alt="g_r=\gamma\frac{M}{r^2}\ \leftrightarrow r=\sqrt[]{\frac{\gamma M}{g_r}}"/><br/> <div>Yksikkötarkastelu</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft%5Br%5Cright%5D%3D%5Csqrt%5B%5D%7B%5Cfrac%7B%5Cleft%5B%5Cgamma%5Cright%5D%5Cleft%5BM%5Cright%5D%7D%7B%5Cleft%5Bg%5Cright%5D%7D%7D%3D%5Csqrt%5B%5D%7B%5Cfrac%7B1%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot1kg%7D%7B1%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%7D%3D%5Csqrt%5B%5D%7B%5Cfrac%7B1%5C%20%5Cfrac%7B%5Cmathrm%7B%5Cfrac%7Bkgm%7D%7Bs%5E2%7D%5Ccdot%20m%5E2%7D%7D%7Bkg%5E2%7D%5Ccdot1kg%7D%7B1%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%7D%3D%5Csqrt%5B%5D%7B%5Cfrac%7B1%5C%20%5Cfrac%7Bm%5E2%7D%7Bs%5E2%7D%7D%7B1%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%7D%3D%5Csqrt%5B%5D%7B1%5C%20m%5E2%7D%3D1m" alt="\left[r\right]=\sqrt[]{\frac{\left[\gamma\right]\left[M\right]}{\left[g\right]}}=\sqrt[]{\frac{1\ \frac{Nm^2}{kg^2}\cdot1kg}{1\ \frac{m}{s^2}}}=\sqrt[]{\frac{1\ \frac{\mathrm{\frac{kgm}{s^2}\cdot m^2}}{kg^2}\cdot1kg}{1\ \frac{m}{s^2}}}=\sqrt[]{\frac{1\ \frac{m^2}{s^2}}{1\ \frac{m}{s^2}}}=\sqrt[]{1\ m^2}=1m"/></div> </div> </div> 2019-04-24T12:10:27+03:00 https://peda.net/id/5b4834fc658 2019-04-24T08:55:11+03:00 3-4 <div>Ganymedes:n kiertoaika on x <div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B%5Cleft(1%7B%2C%7D8d%5Cright)%5E2%7D%7Bx%5E2%7D%3D%5Cfrac%7B%5Cleft(4%5Ccdot10%5E8m%5Cright)%5E3%7D%7B%5Cleft(11%5Ccdot10%5E8m%5Cright)%5E3%7D" alt="\frac{\left(1{,}8d\right)^2}{x^2}=\frac{\left(4\cdot10^8m\right)^3}{\left(11\cdot10^8m\right)^3}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cleft(1%7B%2C%7D8d%5Cright)%5E2%5Ccdot%5Cleft(11%5Ccdot10%5E8m%5Cright)%5E3%3Dx%5E2%5Ccdot%5Cleft(4%5Ccdot10%5E8m%5Cright)%5E3" alt="\left(1{,}8d\right)^2\cdot\left(11\cdot10^8m\right)^3=x^2\cdot\left(4\cdot10^8m\right)^3"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D%5Cfrac%7B%5Cleft(1%7B%2C%7D8d%5Cright)%5E2%5Ccdot%5Cleft(11%5Ccdot10%5E8m%5Cright)%5E3%7D%7B%5Cleft(4%5Ccdot10%5E8m%5Cright)%5E3%7D" alt="x^2=\frac{\left(1{,}8d\right)^2\cdot\left(11\cdot10^8m\right)^3}{\left(4\cdot10^8m\right)^3}"/><br/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%5E2%3D%5Cfrac%7B%5Cleft(1%7B%2C%7D8d%5Cright)%5E2%5Ccdot%5Cleft(11m%5Cright)%5E3%7D%7B%5Cleft(4m%5Cright)%5E3%7D" alt="x^2=\frac{\left(1{,}8d\right)^2\cdot\left(11m\right)^3}{\left(4m\right)^3}"/></div> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Csqrt%5B%5D%7B%5Cfrac%7B%5Cleft(1%7B%2C%7D8d%5Cright)%5E2%5Ccdot%5Cleft(11m%5Cright)%5E3%7D%7B%5Cleft(4m%5Cright)%5E3%7D%7D" alt="x=\sqrt[]{\frac{\left(1{,}8d\right)^2\cdot\left(11m\right)^3}{\left(4m\right)^3}}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D8%7B%2C%7D208...%5Capprox8%7B%2C%7D2d" alt="x=8{,}208...\approx8{,}2d"/><br/> <div><br/> <div>3-6</div> <div>a) Koska Maalla on suurempi massa</div> <div>b) Koska Auringolla suurempi massa</div> <div> </div> <div>3-9</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=m_1%3D420g%3D0%7B%2C%7D42kg" alt="m_1=420g=0{,}42kg"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=m_2%3D0%7B%2C%7D059kg" alt="m_2=0{,}059kg"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r%3D2%7B%2C%7D0m" alt="r=2{,}0m"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F%3D%5Cgamma%5Ccdot%5Cfrac%7Bm_1m_2%7D%7Br%5E2%7D" alt="F=\gamma\cdot\frac{m_1m_2}{r^2}"/> <br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F%3D6%7B%2C%7D67428%5Ccdot10%5E%7B-11%7D%5C%20%5Cfrac%7BNm%5E2%7D%7Bkg%5E2%7D%5Ccdot%5Cfrac%7B0%7B%2C%7D42kg%5Ccdot0%7B%2C%7D059kg%7D%7B%5Cleft(2%7B%2C%7D0m%5Cright)%5E2%7D%3D4%7B%2C%7D134...%5Ccdot10%5E%7B-13%7D%5Capprox4%7B%2C%7D1%5Ccdot10%5E%7B-13%7DN" alt="F=6{,}67428\cdot10^{-11}\ \frac{Nm^2}{kg^2}\cdot\frac{0{,}42kg\cdot0{,}059kg}{\left(2{,}0m\right)^2}=4{,}134...\cdot10^{-13}\approx4{,}1\cdot10^{-13}N"/></div> <div>b)<br/> <span>Koska palloihin vaikuttaa ainoastaan gravitaatiovoima</span><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=F%3Dma%5C%20%5Cleftrightarrow%5C%20a%3D%5Cfrac%7BF%7D%7Bm%7D%3D%5Cfrac%7B4%7B%2C%7D1%5Ccdot10%5E%7B-13%7DN%7D%7B0%7B%2C%7D059g%7D%3D6%7B%2C%7D949...%5Ccdot10%5E%7B-12%7D%3D7%7B%2C%7D0%5Ccdot10%5E%7B-12%7D%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D" alt="F=ma\ \leftrightarrow\ a=\frac{F}{m}=\frac{4{,}1\cdot10^{-13}N}{0{,}059g}=6{,}949...\cdot10^{-12}=7{,}0\cdot10^{-12}\ \frac{m}{s^2}"/><br/> <br/> </div> <div> <div>3-12<br/> <br/> <span>Olkoon Maan ja Auringon välissä kappale, jonka massa on m.Kappale sijaitsee Maan ja Aurigon välissä kohdasssa, jossa vetovoimat ovat yhtä suuret. Olkoon x kysytty etäisyys Maan keskipisteestä lähtien.</span> <div> </div> <div>Maan ja kappaleen välinen gravitaatiovoima on</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F_1%3D%5Cgamma%5Cfrac%7Bm_1m%7D%7Bx%5E2%7D" alt="F_1=\gamma\frac{m_1m}{x^2}"/></div> <div>Auringon ja kappaleen välinen etäisyys on <img src="https://math-demo.abitti.fi/math.svg?latex=r-x" alt="r-x"/> ja gravitaatiovoima on </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F_2%3D%5Cgamma%5C%20%5Cfrac%7Bm_2m%7D%7B%5Cleft(r-x%5Cright)%5E%7B%5E2%7D%7D" alt="F_2=\gamma\ \frac{m_2m}{\left(r-x\right)^{^2}}"/></div> <div>Kappaleeseen vaikuttaa yhtä suuri Maan ja Auringon vetovoima eli <img src="https://math-demo.abitti.fi/math.svg?latex=F_1%3DF_2" alt="F_1=F_2"/></div> <div>Ratkaise Etäisyys x yhtälöllä</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cgamma%5Cfrac%7Bm_1m%7D%7Bx%5E2%7D%3D%5Cgamma%5Cfrac%7Bm_2m%7D%7B%5Cleft(r-x%5Cright)%5E%7B%5E2%7D%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A%5Cgamma" alt="\gamma\frac{m_1m}{x^2}=\gamma\frac{m_2m}{\left(r-x\right)^{^2}}\ \ \ \ \ \left|\right|:\gamma"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bm_1m%7D%7Bx%5E2%7D%3D%5Cfrac%7Bm_2m%7D%7B%5Cleft(r-x%5Cright)%5E2%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3Am" alt="\frac{m_1m}{x^2}=\frac{m_2m}{\left(r-x\right)^2}\ \ \ \ \ \left|\right|:m"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7Bm_1%7D%7Bx%5E2%7D%3D%5Cfrac%7Bm_2%7D%7B%5Cleft(r-x%5Cright)%5E2%7D" alt="\frac{m_1}{x^2}=\frac{m_2}{\left(r-x\right)^2}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=x%3D%5Cfrac%7B-%5Cleft(%5Csqrt%5B%5D%7Bm_1m_2%7D-m_1%5Cright)r%7D%7Bm_1-m_2%7D" alt="x=\frac{-\left(\sqrt[]{m_1m_2}-m_1\right)r}{m_1-m_2}"/><br/> <br/> </div> </div> </div> </div> 2019-04-23T08:40:52+03:00 https://peda.net/id/708f65c860d 2019-04-17T11:26:40+03:00 <div> <div>2-3</div> <div>a) Gravitaati voima</div> <div>b) Istuimen ketjujen jännitysvoima, kitka</div> <div>c) Lepokitka</div> <div> </div> <div>2-4</div> <div>a) Tosi</div> <div>b) Epätosi</div> <div>c) Epätosi</div> <div> </div> <div>2-5</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3D75%5C%20%5Cfrac%7Bkm%7D%7Bh%7D" alt="v=75\ \frac{km}{h}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=r_1%3D320m%3D0%7B%2C%7D32km" alt="r_1=320m=0{,}32km"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=r_2%3D550m%3D0%7B%2C%7D55km" alt="r_2=550m=0{,}55km"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn_1%7D%3D%5Cfrac%7Bv%5E2%7D%7Br_1%7D%3D%5Cfrac%7B75%5C%20%5Cfrac%7Bkm%7D%7Bh%7D%7D%7B0%7B%2C%7D32%5C%20km%7D%3D17578%7B%2C%7D125" alt="a_{n_1}=\frac{v^2}{r_1}=\frac{75\ \frac{km}{h}}{0{,}32\ km}=17578{,}125"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=a_%7Bn_2%7D%3D%5Cfrac%7Bv%5E2%7D%7Br_2%7D%3D%5Cfrac%7B75%5C%20%5Cfrac%7Bkm%7D%7Bh%7D%7D%7B0%7B%2C%7D55%5C%20km%7D%3D10227%7B%2C%7D27273%5C%20" alt="a_{n_2}=\frac{v^2}{r_2}=\frac{75\ \frac{km}{h}}{0{,}55\ km}=10227{,}27273\ "/></div> <div>V: A:ssa on suurempi<br/> <div><br/> <div>2-6</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=m%3D1200%5C%20kg" alt="m=1200\ kg"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=r%3D22m%3D0%7B%2C%7D022km" alt="r=22m=0{,}022km"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3D18%5C%20%5Cfrac%7Bkm%7D%7Bh%7D%3D5%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=18\ \frac{km}{h}=5\ \frac{m}{s}"/></div> <div> <div>a) </div> <div>Koska ilmavastus on pieni, autoon vaikuttava kokonaisvoima on <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3D%5Coverline%7BG%7D%2B%5Coverline%7BN%7D%2B%5Coverline%7BF%7D_%7B%5Cmu%7D" alt="\Sigma\overline{F}=\overline{G}+\overline{N}+\overline{F}_{\mu}"/>. Koska pinta on vaakasuora, pinnan tukivoima <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BN%7D" alt="\overline{N}"/> ja autoon kohdistuva paino<img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BG%7D" alt="\overline{G}"/> ovat yhtä suuret ja vastakkaisuunatiset, joten ne kumoavat toisensa. Kitka<span> </span><img src="https://math-demo.abitti.fi/math.svg?latex=F_%7B%5Cmu%7D%3D%5Cmu%20N%3D%5Cmu%20mg" alt="F_{\mu}=\mu N=\mu mg"/> pitää auton ympyräradalla. Newtonin II lain mukaan jään pinnan suunnassa on </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D_n" alt="\Sigma\overline{F}=m\overline{a}_n"/> eli <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BF%7D_%7B%5Cmu%7D%3Dm%5Coverline%7Ba%7D_n" alt="\overline{F}_{\mu}=m\overline{a}_n"/>. Koska sekä kitka että normaalikiihtyvyys suuntautuvat kohti ympyräradan keskipistettä, yhtälöstä <img src="https://math-demo.abitti.fi/math.svg?latex=F_%7B%5Cmu%7D%3Dma_n" alt="F_{\mu}=ma_n"/>eli <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cmu%20mg%3Dm%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="\mu mg=m\frac{v^2}{r}"/> saadaan kitkakertoimeksi</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cmu%3D%5Cfrac%7Bv%5E2%7D%7Brg%7D%3D%5Cfrac%7B%5Cleft(5%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Cright)%5E2%7D%7B22m%5Ccdot9%7B%2C%7D81%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%3D0%7B%2C%7D1158372718" alt="\mu=\frac{v^2}{rg}=\frac{\left(5\ \frac{m}{s}\right)^2}{22m\cdot9{,}81\ \frac{m}{s^2}}=0{,}1158372718"/></div> </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=F_%7B%5Cmu%7D%3D%5Cmu%20N%3D%5Cmu%20mg%3D0%7B%2C%7D1158372718%5Ccdot1200kg%5Ccdot9%7B%2C%7D81%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%3D1363%7B%2C%7D636364...%5Capprox1400N%3D1%7B%2C%7D4kN" alt="F_{\mu}=\mu N=\mu mg=0{,}1158372718\cdot1200kg\cdot9{,}81\ \frac{m}{s^2}=1363{,}636364...\approx1400N=1{,}4kN"/></div> <div>b) </div> <div>Pysyy, koska saattu kitka on suurempi kuin 1,4kN</div> <div><br/> <div>2-9</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=d%3D1%7B%2C%7D4m%7B%2C%7D%5C%20r%3D0%7B%2C%7D7m" alt="d=1{,}4m{,}\ r=0{,}7m"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=l%3D0%7B%2C%7D9m" alt="l=0{,}9m"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=m%3D0%7B%2C%7D200%5C%20kg" alt="m=0{,}200\ kg"/><br/> <a href="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-2/2-9-png#top" title="2-9.PNG"><img src="https://peda.net/p/kirin_porsti/fysikka/fy5s/teht%C3%A4v%C3%A4t/kpl-2/2-9-png:file/photo/4154a5c1d5bd3d5a0ff5908ee7273324cd8a0f1f/2-9.PNG" alt="" title="2-9.PNG" class="inline" loading="lazy"/></a><br/> <div>a)</div> <div>Oletetaan ilmanvastus pieneksi.</div> <div>Koneeseen vaikuttavat voimat ovat langan jännitysvoima <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BT%7D" alt="\overline{T}"/>ja paino <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BG%7D" alt="\overline{G}"/></div> <div>Langan jännitysvoiman vaakasuuntaan komponentti itä koneen ympyräradalla ja antaa sille normaalikiihtyvyyden <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7Ba%7D_n" alt="\overline{a}_n"/></div> <div>b) </div> <div>Newtonin II-lain mukaan <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D_n" alt="\Sigma\overline{F}=m\overline{a}_n"/></div> <div>y-suunnassa: <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3D%5Coverline%7B0%7D" alt="\Sigma\overline{F}=\overline{0}"/>eli </div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BG%7D%2B%5Coverline%7BT%7D_y%3D%5Coverline%7B0%7D" alt="\overline{G}+\overline{T}_y=\overline{0}"/> <div>Paino ja langan jännitysvoima pusyusiinyaonen kompenentti ovat yhtä suuret ja vastakkaissuuntaiset, joten ne kumoavat toisensa. </div> <div>x-suunnassa: <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BT%7D_x%3Dm%5Coverline%7Ba%7D_n" alt="\overline{T}_x=m\overline{a}_n"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=T_x%3Dma_n" alt="T_x=ma_n"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=T_x%3Dm%5Cfrac%7Bv%5E2%7D%7Br%7D" alt="T_x=m\frac{v^2}{r}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=rT_x%3Dmv%5E2" alt="rT_x=mv^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7BrT_x%7D%7Bm%7D%3Dv%5E2" alt="\frac{rT_x}{m}=v^2"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=v%3D%5Cpm%5Csqrt%5B%5D%7B%5Cfrac%7BrT_x%7D%7Bm%7D%7D" alt="v=\pm\sqrt[]{\frac{rT_x}{m}}"/>(hyl. neg. tulos)</div> <div>Ratkaistaan <img src="https://math-demo.abitti.fi/math.svg?latex=T_x" alt="T_x"/>suorakulmaisen kolmion trigonometrian avulla. </div> <div>Ratkaistaan kulma α</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Ccos%5Calpha%3D%5Cfrac%7Br%7D%7Bl%7D%3D%5Cfrac%7B0%7B%2C%7D7m%7D%7B0%7B%2C%7D9m%7D" alt="\cos\alpha=\frac{r}{l}=\frac{0{,}7m}{0{,}9m}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Calpha%3D38%7B%2C%7D94244...%5Capprox38%7B%2C%7D9424%C2%B0" alt="\alpha=38{,}94244...\approx38{,}9424°"/></div> <div>Suunnat huomioiden saadaan</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=G%3DT_y" alt="G=T_y"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Ctan%5Calpha%3D%5Cfrac%7BG%7D%7BT_x%7D" alt="\tan\alpha=\frac{G}{T_x}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=T_x%3D%5Cfrac%7BG%7D%7B%5Ctan%5Calpha%7D%3D%5Cfrac%7Bmg%7D%7B%5Ctan%5Calpha%7D%3D%5Cfrac%7B0%7B%2C%7D200%5C%20kg%5Ccdot9%7B%2C%7D81%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%7D%7B%5Ctan%5C%2038%7B%2C%7D9424%7D%3D2%7B%2C%7D427854...%5Capprox2%7B%2C%7D42785N" alt="T_x=\frac{G}{\tan\alpha}=\frac{mg}{\tan\alpha}=\frac{0{,}200\ kg\cdot9{,}81\ \frac{m}{s^2}}{\tan\ 38{,}9424}=2{,}427854...\approx2{,}42785N"/><br/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=v%3D%5Csqrt%5B%5D%7B%5Cfrac%7BrT_x%7D%7Bm%7D%7D%3D%5Csqrt%5B%5D%7B%5Cfrac%7B0%7B%2C%7D7%5C%20m%5Ccdot2%7B%2C%7D42785%5C%20N%7D%7B0%7B%2C%7D200%5C%20kg%7D%7D%3D2%7B%2C%7D9%5C%20%5Cfrac%7Bm%7D%7Bs%7D" alt="v=\sqrt[]{\frac{rT_x}{m}}=\sqrt[]{\frac{0{,}7\ m\cdot2{,}42785\ N}{0{,}200\ kg}}=2{,}9\ \frac{m}{s}"/><br/> <br/> </div> </div> <div> <div> <div>2-10</div> <div>Laskijan liikeyhtälö on <img src="https://math-demo.abitti.fi/math.svg?latex=%5CSigma%5Coverline%7BF%7D%3Dm%5Coverline%7Ba%7D_n" alt="\Sigma\overline{F}=m\overline{a}_n"/> eli <img src="https://math-demo.abitti.fi/math.svg?latex=%5Coverline%7BN%7D%2B%5Coverline%7BG%7D%3Dm%5Coverline%7Ba%7D_n" alt="\overline{N}+\overline{G}=m\overline{a}_n"/></div> Kun suunta ylös on positiivinen, skalaariyhtälöstä <img src="https://math-demo.abitti.fi/math.svg?latex=N-G%3D%5Cfrac%7Bmv%5E2%7D%7Br%7D" alt="N-G=\frac{mv^2}{r}"/> rinteen tukivoiman suurudeksi alimmassa kohdassa saadaan</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=N%3D%5Cfrac%7Bmv%5E2%7D%7Br%7D%2BG%3D%5Cfrac%7Bmv%5E2%7D%7Br%7D%2Bmg%3D%5Cfrac%7B82%5C%20kg%5Ccdot%5Cleft(%5Cfrac%7B45%7D%7B3%7B%2C%7D6%7D%5C%20%5Cfrac%7Bm%7D%7Bs%7D%5Cright)%5E2%7D%7B22m%7D%2B82kg%5Ccdot9%7B%2C%7D81%5C%20%5Cfrac%7Bm%7D%7Bs%5E2%7D%3D1386%7B%2C%7D806...%5Capprox1400N%3D1%7B%2C%7D4kN" alt="N=\frac{mv^2}{r}+G=\frac{mv^2}{r}+mg=\frac{82\ kg\cdot\left(\frac{45}{3{,}6}\ \frac{m}{s}\right)^2}{22m}+82kg\cdot9{,}81\ \frac{m}{s^2}=1386{,}806...\approx1400N=1{,}4kN"/></div> Rinteestä laskijan jalkoihin kohdistuvan voiman suuruus on yhtä suuri kuin rinteen tukivoiman suuruus eli 1,4kN</div> </div> 2019-04-17T09:38:41+03:00 https://peda.net/id/9daa0d8c5fa 2019-04-15T20:48:02+03:00 <span>Esimerkki </span> <div>a) Muunna asteiksi 1,2 rad </div> <div>b) Muunna radiaaneiksi 185°</div> <div> </div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=2%5Cpi%5Cmathrm%7B%5C%20rad%7D%3D360%C2%B0" alt="2\pi\mathrm{\ rad}=360°"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cpi%5C%20%5Cmathrm%7Brad%7D%3D180%C2%B0" alt="\pi\ \mathrm{rad}=180°"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=1%5Cmathrm%7B%5C%20rad%7D%3D%5Cfrac%7B180%C2%B0%7D%7B%5Cpi%7D" alt="1\mathrm{\ rad}=\frac{180°}{\pi}"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=1%7B%2C%7D2%5C%20%5Cmathrm%7Brad%7D%3D1%7B%2C%7D2%5Ccdot%5Cfrac%7B180%C2%B0%7D%7B%5Cpi%7D%5Capprox69%C2%B0" alt="1{,}2\ \mathrm{rad}=1{,}2\cdot\frac{180°}{\pi}\approx69°"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=180%C2%B0%3D%5Cpi%5C%20%5Cmathrm%7Brad%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%3A180%C2%B0" alt="180°=\pi\ \mathrm{rad}\ \ \ \ \ \left|\right|:180°"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=1%C2%B0%3D%5Cfrac%7B%5Cpi%7D%7B180%7D%5C%20%5Cmathrm%7Brad%7D%5C%20%5C%20%5C%20%5C%20%5C%20%5Cleft%7C%5Cright%7C%5Ccdot185%C2%B0" alt="1°=\frac{\pi}{180}\ \mathrm{rad}\ \ \ \ \ \left|\right|\cdot185°"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=185%C2%B0%3D185%5Ccdot%5Cfrac%7B%5Cpi%7D%7B180%7D%5Capprox3%7B%2C%7D2%5C%20%5Cmathrm%7Brad%7D" alt="185°=185\cdot\frac{\pi}{180}\approx3{,}2\ \mathrm{rad}"/> <div> </div> <div>1-1</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=d%3D8%7B%2C%7D5cm" alt="d=8{,}5cm"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=r%3D4%7B%2C%7D25%5C%20cm" alt="r=4{,}25\ cm"/> <div><img src="https://math-demo.abitti.fi/math.svg?latex=P%3D2%5Cpi%20r%3D2%5Ccdot%5Cpi%5Ccdot4%7B%2C%7D25cm%3D%5Cfrac%7B17%7D%7B2%7D%5Cpi" alt="P=2\pi r=2\cdot\pi\cdot4{,}25cm=\frac{17}{2}\pi"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=15km%3D15000m%3D1500000cm" alt="15km=15000m=1500000cm"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B1500000%5C%20cm%7D%7B%5Cfrac%7B17%7D%7B2%7D%5Cpi%5C%20cm%7D%3D56172%7B%2C%7D33...%5Capprox56000%5C%20%5Cmathrm%7Bkierrosta%7D" alt="\frac{1500000\ cm}{\frac{17}{2}\pi\ cm}=56172{,}33...\approx56000\ \mathrm{kierrosta}"/></div> <div> </div> <div>1-2</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=0%7B%2C%7D7%5Ccdot%5Cfrac%7B180%7D%7B%5Cpi%7D%3D40%7B%2C%7D10704...%5Capprox40%7B%2C%7D11%C2%B0" alt="0{,}7\cdot\frac{180}{\pi}=40{,}10704...\approx40{,}11°"/><br/> <img src="https://math-demo.abitti.fi/math.svg?latex=6%7B%2C%7D0%5Ccdot%5Cfrac%7B180%7D%7B%5Cpi%7D%3D343%7B%2C%7D7746...%5Capprox343%7B%2C%7D77%C2%B0" alt="6{,}0\cdot\frac{180}{\pi}=343{,}7746...\approx343{,}77°"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=45%5Ccdot%5Cfrac%7B%5Cpi%7D%7B180%7D%3D0%7B%2C%7D785...%5Capprox0%7B%2C%7D79%5C%20%5Cmathrm%7Brad%7D" alt="45\cdot\frac{\pi}{180}=0{,}785...\approx0{,}79\ \mathrm{rad}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=750%5Ccdot%5Cfrac%7B%5Cpi%7D%7B180%7D%3D13%7B%2C%7D0899...%5Capprox13%5C%20%5Cmathrm%7Brad%7D" alt="750\cdot\frac{\pi}{180}=13{,}0899...\approx13\ \mathrm{rad}"/><br/> <div><br/> <div>1-5</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cfrac%7B60s%7D%7B14%5C%20000%5C%20rpm%7D%3D4%7B%2C%7D285...%5Ccdot10%5E%7B-3%7Ds%5Capprox4%7B%2C%7D3%5C%20ms" alt="\frac{60s}{14\ 000\ rpm}=4{,}285...\cdot10^{-3}s\approx4{,}3\ ms"/></div> </div> <div><br/> <div>1-6</div> <div>a) Kiertokulma on <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cvarphi%3D90%C2%B0%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cleft(rad%5Cright)" alt="\varphi=90°=\frac{\pi}{2}\left(rad\right)"/></div> <div>Kiertokulma on ympyrän kaaren pituuden ja säteen suhde eli <img src="https://math-demo.abitti.fi/math.svg?latex=%5Cvarphi%3D%5Cfrac%7Bs%7D%7Br%7D" alt="\varphi=\frac{s}{r}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%3Dr%5Cvarphi%3D15m%5Ccdot%5Cfrac%7B%5Cpi%7D%7B2%7D%5Capprox23%7B%2C%7D562%5Capprox24%5C%20m" alt="s=r\varphi=15m\cdot\frac{\pi}{2}\approx23{,}562\approx24\ m"/></div> b) <div><img src="https://math-demo.abitti.fi/math.svg?latex=s%3Dr%5Cvarphi%3D7%7B%2C%7D5m%5Ccdot%5Cfrac%7B%5Cpi%7D%7B2%7D%5Capprox11%7B%2C%7D78...%5Capprox12%5C%20m" alt="s=r\varphi=7{,}5m\cdot\frac{\pi}{2}\approx11{,}78...\approx12\ m"/></div> <div> </div> <div>1-7</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=N%3D1" alt="N=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=%5CDelta%20t%3D10%7B%2C%7D8s" alt="\Delta t=10{,}8s"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=n%3D%5Cfrac%7BN%7D%7B%5CDelta%20t%7D%3D0%7B%2C%7D09259...%5Capprox0%7B%2C%7D0926%5Cfrac%7Br%7D%7Bs%7D" alt="n=\frac{N}{\Delta t}=0{,}09259...\approx0{,}0926\frac{r}{s}"/></div> <div>b) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5CDelta%20t%3D2%5C%20%5Cmin%3D120%5C%20s" alt="\Delta t=2\ \min=120\ s"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Comega%3D2%5Cpi%20n%3D2%5Cpi%5Ccdot0%7B%2C%7D09256%5C%20%5Cfrac%7Br%7D%7Bs%7D%3D0%7B%2C%7D581776...%5Capprox0%7B%2C%7D582%5C%20%5Cmathrm%7B%5Cfrac%7Brad%7D%7Bs%7D%7D" alt="\omega=2\pi n=2\pi\cdot0{,}09256\ \frac{r}{s}=0{,}581776...\approx0{,}582\ \mathrm{\frac{rad}{s}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Cvarphi%3D%5Comega%5CDelta%20t%3D0%7B%2C%7D5817%5C%20%5Cfrac%7Brad%7D%7Bs%7D%5Ccdot120s%3D69%7B%2C%7D813...%5Capprox70%5Cmathrm%7B%5C%20rad%7D" alt="\varphi=\omega\Delta t=0{,}5817\ \frac{rad}{s}\cdot120s=69{,}813...\approx70\mathrm{\ rad}"/><br/> <div> </div> <div>1-10</div> <div>a)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5CDelta%20t%3D7%7B%2C%7D0%5C%20s" alt="\Delta t=7{,}0\ s"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=N%3D1" alt="N=1"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D%5Cfrac%7B1%7D%7B7%7B%2C%7D0s%7D%3D0%7B%2C%7D1428...%5Capprox0%7B%2C%7D143%5C%20%5Cfrac%7Br%7D%7Bs%7D" alt="n=\frac{1}{7{,}0s}=0{,}1428...\approx0{,}143\ \frac{r}{s}"/></div> <div>b)</div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Comega%3D2%5Cpi%20n%3D2%5Cpi%5Ccdot0%7B%2C%7D1428%5C%20%5Cfrac%7Br%7D%7Bs%7D%3D0%7B%2C%7D8975...%5Capprox0%7B%2C%7D898%5C%20%5Cmathrm%7B%5Cfrac%7Brad%7D%7Bs%7D%7D" alt="\omega=2\pi n=2\pi\cdot0{,}1428\ \frac{r}{s}=0{,}8975...\approx0{,}898\ \mathrm{\frac{rad}{s}}"/></div> <div> </div> <div>1-12</div> <div>a)</div> <div>Kyllä, koska niiden liikeistä on mahdollista saada lireaarisen kuvaajan.</div> <div>b) </div> <div>A, koska sen kuvaajan kuvaaja on jyrkempi (n. 13 rad/s)</div> <div>c) </div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Comega_A%3D13%7B%2C%7D3333...%5Capprox13%7B%2C%7D3%5C%20%5Cmathrm%7B%5Cfrac%7Brad%7D%7Bs%7D%7D" alt="\omega_A=13{,}3333...\approx13{,}3\ \mathrm{\frac{rad}{s}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=%5Comega_B%3D5%5C%20%5Cmathrm%7B%5Cfrac%7Brad%7D%7Bs%7D%7D" alt="\omega_B=5\ \mathrm{\frac{rad}{s}}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=n_A%3D%5Cfrac%7B%5Comega_A%7D%7B2%5Cpi%7D%3D2%7B%2C%7D12206...%5Capprox2%7B%2C%7D12%5C%20%5Cfrac%7Br%7D%7Bs%7D" alt="n_A=\frac{\omega_A}{2\pi}=2{,}12206...\approx2{,}12\ \frac{r}{s}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=n_B%3D%5Cfrac%7B%5Comega_B%7D%7B2%5Cpi%7D%3D0%7B%2C%7D79577..%5Capprox0%7B%2C%7D80%5C%20%5Cfrac%7Br%7D%7Bs%7D" alt="n_B=\frac{\omega_B}{2\pi}=0{,}79577..\approx0{,}80\ \frac{r}{s}"/></div> <img src="https://math-demo.abitti.fi/math.svg?latex=n%3D%5Cfrac%7BN%7D%7Bt%7D%5C%20%5Cleftrightarrow%5C%20t%3D%5Cfrac%7BN%7D%7Bn%7D" alt="n=\frac{N}{t}\ \leftrightarrow\ t=\frac{N}{n}"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=t_A%3D%5Cfrac%7B3%7D%7B2%7B%2C%7D122%5C%20%5Cfrac%7Br%7D%7Bs%7D%7D%3D1%7B%2C%7D4137...%5Capprox1%7B%2C%7D41%5C%20s" alt="t_A=\frac{3}{2{,}122\ \frac{r}{s}}=1{,}4137...\approx1{,}41\ s"/></div> <div><img src="https://math-demo.abitti.fi/math.svg?latex=t_B%3D%5Cfrac%7B3%7D%7B0%7B%2C%7D79577%5C%20%5Cfrac%7Br%7D%7Bs%7D%7D%3D3%7B%2C%7D76993...%5Capprox3%7B%2C%7D77%5C%20s" alt="t_B=\frac{3}{0{,}79577\ \frac{r}{s}}=3{,}76993...\approx3{,}77\ s"/></div> 2019-04-15T20:48:02+03:00