Person sits on a frictionless stool that is free to try ate but is initially at rest. The person is holding a bicycle wheel (I = 3 kg*m2) that is rotating at 8 Ree/s in a clockwise direction as viewed from above, and the moment of inertia of the person-wheel-stool system is 9 kg *m2.
1. What is the magnitude of the angular velocity of the PWS?
2. What is the magnitude of the angular momentum of the PES?
3. What is the magnitude of the angular acceleration of the person-wheel-stool system?
To answer these questions, we'll use the principles of conservation of angular momentum.
1. To find the magnitude of the angular velocity of the person-wheel-stool (PWS) system, we can use the conservation of angular momentum equation:
Initial angular momentum = Final angular momentum
Initially, the person-wheel-stool system is at rest, so the initial angular momentum is zero. The moment of inertia of the system (I_sys) is given as 9 kg*m^2. The person starts rotating the bicycle wheel, creating angular momentum. The bicycle wheel has a moment of inertia (I_wheel) of 3 kg*m^2 and is rotating at 8 rev/s (or 8 * 2π rad/s) in a clockwise direction.
The final angular momentum of the system is the sum of the angular momentum of the person (L_person) and the angular momentum of the wheel (L_wheel):
L_final = L_person + L_wheel
L_final = (I_person * ω_person) + (I_wheel * ω_wheel)
Since the person is sitting on a frictionless stool and the system is closed with no external torques acting on it, the angular momentum is conserved. Therefore, the final angular momentum is equal to the initial angular momentum (zero).
0 = (I_person * ω_person) + (I_wheel * ω_wheel)
Rearranging the equation, we can solve for ω_person:
ω_person = -((I_wheel * ω_wheel) / I_person)
Substituting the given values:
ω_person = -((3 kg*m^2) * (8 * 2π rad/s)) / 9 kg*m^2
Calculating this will give you the magnitude of the angular velocity of the PWS system.
2. The angular momentum (L) of an object is given by the equation:
L = I * ω
To find the magnitude of the angular momentum of the PWS system, we can use the equation:
L_sys = I_sys * ω_sys
Substituting the given values:
L_sys = 9 kg*m^2 * ω_sys
ω_sys is the angular velocity of the person-wheel-stool system, which we calculated in the previous step.
Calculating this will give you the magnitude of the angular momentum of the PES system.
3. To find the magnitude of the angular acceleration of the person-wheel-stool system, we'll need to differentiate the angular velocity with respect to time:
α_sys = dω_sys / dt
Since the person starts rotating the wheel, the angular acceleration is not zero. However, the given information does not provide any details about the applied torque or the time involved. Therefore, without additional information, it is not possible to determine the exact magnitude of the angular acceleration.