Opgave 2.3, 2.5 og 2.7


Opgave 2.3

Inertimoment
LaTex: \begin{eqnarray} r_{hjul} &=& \frac{1m}{2} = 0,5m\\ m_{hjul} &=& 100 kg\\ I_{hjul} &=& m_{hjul}\cdot r_{hjul}^2 = 25 kg\cdot m^2\\ \end{eqnarray}
Vinkel hastighed
LaTex: \begin{eqnarray} m &=& 1200 kg\\ v &=& 100 \frac{km}{h} = 27,78 \frac{m}{s}\\ E_{kin} &=& \frac{1}{2}\cdot m\cdot v^2 = 462,96 kJ\\ 0,75\cdot E_{kin} &=& \frac{1}{2}\cdot I_{hjul}\cdot \omega^2\\ \omega &=&  \sqrt{\frac{0,75\cdot E_{kin}}{\frac{1}{2}\cdot I_{hjul}}} = 166,67s^{-1}\\ \end{eqnarray}
Omdrejninger pr. min
LaTex: \begin{eqnarray} T &=& \frac{2\cdot\pi\cdot 60}{\omega} = 2,26\\ \end{eqnarray}

Opgave 2.5

Vinkel hastighed
LaTex: \begin{eqnarray} v_p &=& 200 \frac{m}{s}\\ m_p &=& 10g\\ m_s &=& 2 kg\\ l_s &=& 0,5 m\\ I_0 &=&\frac{1}{12}\cdot m_s\cdot l_s = 0,041666667 kg\cdot m^2\\ I &=& (\frac{l_s}{2})^2\cdot m_s + I_0 = 0,16666667 kg\cdot m^2\\ p &=& m_p\cdot v_p = 2 Ns\\ L &=& p\cdot l_s = 1 Nms\\ \omega &=& \frac{L}{I} = 6 s^{-1}\\ \end{eqnarray}
Udslagsstørrelse
LaTex: \begin{eqnarray} E_{rot} &=& \frac{1}{2}\cdot I\cdot\omega^2 = 3J\\ E_{pot} &=& m\cdot g\cdot h = E_{rot}\\ h &=& \frac{E_{rot}}{m\cdot g} = 15,3 cm\\ \phi &=& 90 - sin^{-1}\left(\frac{sin(90)}{\frac{l_s}{2}}\cdot \left( \frac{l_s}{2}-h \right)\right) = 67,2^\circ C\\ \end{eqnarray}

Opgave 2.7

Bestem inertimoment
LaTex: \begin{eqnarray} m_{pige} &=& 50 kg\\ r &=& 2m\\ I_{karusel} &=& 100 kg\cdot m^2\\ T_{peri} &=& 6 s\\ m &=& \frac{I_{karusel}\cdot 2}{r^2} = 50 kg\\ I_{peri} &=& m_{pige}\cdot r^2 + I_{karusel} = 300 kg\cdot m^2\\ I_{cent} &=& m_{pige}\cdot \left(\frac{r}{2}\right)^2 + I_{karusel} = 150 kg\cdot m^2\\ \end{eqnarray}
Omløbstiden
LaTex: \begin{eqnarray} \omega_{peri} &=& \frac{2\cdot \pi}{T_{peri}} = 1,047 s^{-1}\\ L &=& I_{peri}\cdot\omega_{peri} = I_{cent}\cdot\omega_{cent}\\ \omega_{cent} &=& \frac{I_{peri}\cdot\omega_{peri}}{I_{cent}} = 2,094 s^{-1}\\ T_{cent} &=& \frac{2\cdot\pi}{\omega_{cent}} = 3s\\ \end{eqnarray}
Udført arbejde
LaTex: \begin{eqnarray} E_{rot_{peri}} &=& \frac{1}{2}\cdot I_{peri}\cdot\omega_{peri}^2 = 164,43 J\\ E_{rot_{cent}} &=& \frac{1}{2}\cdot I_{cent}\cdot\omega_{cent}^2 = 328,86 J\\ A &=& E_{rot_{cent}} - E_{rot_{peri}} = 164,43J\\ \end{eqnarray}