Question

A turntable has a radius R and mass M (considered as a disk) and is rotating at an angular velocity W0 about a frictionless vertical axis. A piece of clay is tossed onto the turntable and sticks d from the rotational axis. The clay hits with horizontal velocity component vc at right angle to the turntable’s radius, and in a direction that opposes the rotation. After the clay lands, the turntable has slowed to angular velocity W1. A) Find the mass of the clay mc. B) How much energy is lost during the collision process (i.e., due to dissipation and deformation of the clay)?

Answer 1

A)

I_t = Moment of inertia of turntable = (0.5) M R²

W_o = initial angular velocity of turntable

I_c = moment of inertia of clay = m_c d²

W₁ = final angular velocity

using conservation of angular momentum

I_t W_o - m_c v_c d = (I_t + I_c) W₁

I_t W_o - m_c v_c d = (I_t + m_c d²) W₁

I_t (W_o + W₁) = m_c v_c d + m_c d² W₁

(0.5) M R² (W_o + W₁) = m_c v_c d + m_c d² W₁

m_c = (0.5) M R² (W_o + W₁) /(v_c d + d² W₁)

b)

Energy lost is given as

E = (0.5) (I_t W²_o + m_c v²_c - (I_t + I_c) W²₁)

E = (0.5) ((0.5) M R² W²_o + m_c v²_c - ((0.5) M R² + m_cd²) W²₁)