A horizontal disk with a radius of 0.5 meters and a moment of inertia of 2 kg·m² is spinning freely at 10 revolutions per minute clockwise. A student drops a small object with a mass of 0.1 kg onto the disk at a distance of 0.3 meters from the center of the disk. The object sticks to the disk upon impact.
a) What is the initial angular momentum of the disk?
b) What is the final angular velocity of the disk and the object together?
c) Is angular momentum conserved in this situation? Justify your answer.
a) The initial angular momentum (L_initial) of the disk is given by the equation:
L_initial = I * ω_initial
where I is the moment of inertia of the disk and ω_initial is the initial angular velocity.
Given: I = 2 kg·m² ω_initial = (10 rev/min) * (2π rad/rev) * (1 min/60 s) = 1.05 rad/s
Substituting the values into the equation, we can calculate the initial angular momentum:
L_initial = (2 kg·m²) * (1.05 rad/s) = 2.1 kg·m²/s
Therefore, the initial angular momentum of the disk is 2.1 kg·m²/s.
b) After the object sticks to the disk, the total moment of inertia (I_final) of the disk and the object can be calculated by adding their respective moments of inertia:
I_final = I_disk + I_object
Given: I_disk = 2 kg·m² (from part a) I_object = m_object * r² m_object = 0.1 kg (given) r = 0.3 m (given)
Substituting the values, we can calculate the moment of inertia of the object:
I_object = (0.1 kg) * (0.3 m)² = 0.009 kg·m²
Therefore, the final moment of inertia of the system is:
I_final = 2 kg·m² + 0.009 kg·m² = 2.009 kg·m²
Now, to find the final angular velocity (ω_final), we can use the conservation of angular momentum:
L_initial = L_final
L_initial = I_initial * ω_initial L_final = I_final * ω_final
Therefore,
I_initial * ω_initial = I_final * ω_final
Substituting the values,
(2 kg·m²) * (1.05 rad/s) = (2.009 kg·m²) * ω_final
Solving for ω_final,
ω_final = [(2 kg·m²) * (1.05 rad/s)] / (2.009 kg·m²) = 1.1005 rad/s
Hence, the final angular velocity of the disk and the object together is approximately 1.1005 rad/s.
c) Angular momentum is conserved in this situation. This is justified by the principle of conservation of angular momentum, which states that the total angular momentum of an isolated system remains constant unless acted upon by external torques. In this case, no external torques are acting on the system, so the initial angular momentum of the disk is equal to the final angular momentum of the disk and object combined. Therefore, angular momentum is conserved.