Drums

Classical Mechanics Level 4

A drum has a radius of $0.40\mbox{ m}$ and a moment of inertia of $4.5\mbox{ kg m}^2$ . The frictional torque of the drum axle is $3.0\mbox{ N m}$ . A $15\mbox{ m}$ length of rope is wound around the rim. The drum is initially at rest. A constant force is applied to the free end of the rope until the rope is completely unwound and slips off. At that instant, the angular velocity of the drum is $13\mbox{ rad/s}$ . The drum then decelerates and comes to a halt. The constant force applied to the rope is closest to

33 7.5 18 27 14

1 solution

Gandoff Tan
Dec 4, 2019

$\begin{aligned} F&=\frac{\tau_F}{r}\\ &=\frac{\tau-\tau_{\text{friction}}}{r}\\ &=\frac{\frac{\Delta L}{\Delta t}}{r}-\frac{\tau_{\text{friction}}}{r}\\ &=\frac{I\Delta\omega}{r\Delta t}-\frac{\tau_{\text{friction}}}{r}\\ &=\frac{I\Delta\omega}{r\frac{2ℓ}{r\Delta\omega}}-\frac{\tau_{\text{friction}}}{r}\\ &=\frac{I(\Delta\omega)^2}{2ℓ}-\frac{\tau_{\text{friction}}}{r}\\ &=\frac{\left(\SI{4.5}{\kg\m^2}\right)\left(\SI{13}{\s^{-1}}\right)^2}{2(\SI{15}{\m})}-\frac{\SI{-3.0}{\N\m}}{\SI{0.40}{\m}}\\ &=\SI{25.35}{\N}+\SI{7.5}{\N}\\ &=\SI{32.85}{\N}\\ &≈\boxed{\SI{33}{\N}} \end{aligned}$

Drums

1 solution

0 pending reports