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A track is mounted on a large wheel that is free to turn with negligible friction about a vertical axis. Consider a toy train of mass M placed on the track with the system initially at rest, the train’s electrical power is turned on. The train reaches speed v with respect to the track. What is the wheel’s angular speed if its mass is m and its radius is r? (Treat it as a hoop, and neglect the mass of the spokes and hub).

a) \(\begin{array}{l}\frac{v}{(\frac{m}{M}+1)R}\end{array} \) b) \(\begin{array}{l}\frac{v}{(\frac{m}{M}+2)R}\end{array} \) c) \(\begin{array}{l}\frac{v}{(\frac{M}{m}-1)R}\end{array} \) d) \(\begin{array}{l}\frac{v}{(\frac{m}{M}-2)R}\end{array} \)   Solution: a) \(\begin{array}{l}\frac{v}{(\frac{m}{M}+1)R}\end{array} \)  

Calculate the total energy UB and the energy density Up of the magnetic field stored in a solenoid 0.5 m long, having 5000 turns and a current 10A. (radius of the solenoid is 4 cm).

Solution: \(\begin{array}{l}B=\frac{\mu_{0}NI}{l}\end{array} \) \(\begin{array}{l}B=\frac{4*\pi *10^{-7}*5000*10}{0.5}\end{array} \) \(\begin{array}{l}B=0.04\pi\end{array} \) \(\begin{array}{l}E=\frac{B^{2}}{2\mu_{0}}A_{L}\end{array} \) \(\begin{array}{l}E=\frac{(0.04\pi)^{2}}{2*4\pi *10^{-7}}*\pi (0.04)^{2}*0.5\end{array} \) E=15.7 joules