Ph203 Quiz 9           November 6, 2000

1. An automatic dryer spins wet clothes at an angular speed of 4.6 rad/s. Starting from rest, the drier reaches its operating speed with an average angular acceleration of 3.0 rad/s$^2$. How long does it take the dryer to come up to speed? [4 pts]

   Solution: This is an angular kinematics problem. You know the initial angular speed (0 rad/s) and the final angular speed (4.6 rad/s), so from the definition of average angular acceleration, $\alpha$:

   
   

   $$\alpha = {\Delta \omega \over \Delta t}$$
   
   

   you can determine the spin-up time:

   
   

   $$t = {\Delta \omega \over \alpha} = {4.6 - 0 \over 3.0 }\, \rm s = 1.53\, \rm s$$
   
   

2. A baggage carousel at an airport is rotating with an angular speed of 0.15 rad/s when the baggage begins to be loaded onto it. The moment of inertia of the carousel is 2000 kg m$^2$. Ten pieces of baggage with an average mass of 10 kg each are dropped vertically onto the carousel and come to rest (on the carousel) at a perpendicular distance of 2.0 m from the axis of rotation. (a) Assuming that no net external torque acts on the system of carousel and baggage, find the final angular speed. [4 pts] (b) In reality, the angular speed of the baggage carousel does not change. How can this be so? Explain in a sentence or two. [2 pts]

   Solution: (a) This is a conservation of angular momentum problem. The moment of inertia changes, so the angular velocity must change too. The moment of inertia of the carousel alone is $I_C$ and it becomes $I_C + 10 M R^2$ when the 10 bags of mass $M$ are added to it (at a radius $R$ from the rotation axis). The initial angular momentum comes solely from the rotation of the carousel.

   
   

   $$I_C \omega_0 = (I_C + 10 MR^2) \omega_f$$
   
   

   Then

   
   

   $$\omega_f = {I_C \over I_C + 10 MR^2}\, \omega_0 = {2000 \over... ...00}\, 0.15 {\, \rm rad\, s^{-1}} = 0.125 {\, \rm rad\, s^{-1}}$$
   
   

   (b) The angular speed of the baggage carousel does not change because there is a motor powering the carousel, i.e., the motor applies an external torque to keep the angular velocity constant.