A turntable has a radius of 0.80 m and a moment of inertia of 2.0 kg · m².The turntable is rotating with an angular velocity of 1.5 rad/s about a vertical axis though its center on frictionless bearings.A very small 0.40-kg ball is projected horizontally toward the turntable axis with a velocity of 3.0 m/s.The ball is caught by a very small and very light cup-shaped mechanism on the rim of the turntable (see figure).What is the angular velocity of the turntable just after the ball is caught? A)2.1 rad/s B)1.3 rad/s C)0.94 rad/s D)0.75 rad/s E)0.30 rad/s

Question

Quizplus · Accepted Answer

The conservation of angular momentum can be applied to the system of the turntable and the ball. Before the ball is caught, the angular momentum of the system is:
L = Iω = (2.0 kg · m²)(1.5 rad/s) = 3.0 kg · m²/s
After the ball is caught, the turntable will experience a slight increase in its moment of inertia due to the added mass of the ball. However, the effect of this increase is negligible compared to the effect of the ball's angular momentum. The ball's angular momentum is:
L_ball = mvr = (0.40 kg)(0.80 m)(3.0 m/s) = 0.96 kg · m²/s
Since the turntable is initially rotating about a vertical axis through its center, the angular momentum of the ball is initially in the horizontal direction. When the ball is caught by the mechanism on the rim of the turntable, its angular momentum is transfered to the turntable in the vertical direction. Therefore, the final angular momentum of the system is:
L_f = L + L_ball = 3.96 kg · m²/s
The final angular velocity of the turntable can be calculated using the final angular momentum and the moment of inertia of the system:
L_f = I_fω_f
ω_f = L_f / I_f
where I_f = I + mR² is the moment of inertia of the system after the ball is caught, where R is the radius of the turntable. Substituting the given values, we get:
ω_f = (3.96 kg · m²/s) / (2.0 kg · m² + 0.40 kg * (0.80 m)²) = 1.27 rad/s
Therefore, the angular velocity of the turntable just after the ball is caught is 1.3 rad/s (to two significant figures), which corresponds to choice (B).

A Turntable Has a Radius of 0