Auniform cylindrical turntable of radius 1.80 m and mass 29.5 kg rotates counterclockwise in a horizontal plane with an initial angular speed of 4π rad/s. the fixed turntable bearing is frictionless. a lump of clay of mass 2.32 kg and negligible size is dropped onto the turntable from a small distance above it and immediately sticks to the turntable at a point 1.70 m to the east of the axis. (a) find the final angular speed of the clayand turntable.

3.5π rad/s

Explanation:

R = radius of the turntable = 1.80 m

M = mass of the turntable = 29.5 kg

m = mass of the lump of clay = 2.32 kg

r = distance of lump of clay from axis of rotation = 1.70 m

I₀ = Initial moment of inertia of turntable

Initial moment of inertia of turntable is given as

I₀ = (0.5) M R²

w₀ = initial angular speed = 4π rad/s

I = Final moment of inertia of turntable

Final moment of inertia of turntable is given as

I = I₀ + m r² = (0.5) M R² + m r²

w = final angular speed

Using conservation of angular momentum

I₀ w₀ = I w

(0.5) M R² w₀ = ((0.5) M R² + m r²) w

(0.5) (29.5) (1.80)² (4π) = ((0.5) (29.5) (1.80)² + (2.32) (1.70)²) w

w = 3.5π rad/s