A 2.0 kg, 20-cm-diameter turntable rotates at 100 rpm on frictionless bearings. Two 500 g blocks fall from above, hit the turntable simultaneously at opposite ends of a diameter, and stick. What is the turntable's angular velocity, in rpm, just after this event?

검증된 단계별 안내

Step 1: Identify the principle of conservation of angular momentum. Since there are no external torques acting on the system, the total angular momentum before and after the blocks stick to the turntable must be conserved.

Step 2: Calculate the initial angular momentum of the turntable. Use the formula for the moment of inertia of a solid disk:

I = \frac{1}{2} M R^{2}

, where

M

is the mass of the turntable and

R

is its radius. Then multiply the moment of inertia by the initial angular velocity (converted to radians per second) to find the initial angular momentum.

Step 3: Determine the moment of inertia of the system after the blocks stick to the turntable. The blocks can be treated as point masses located at a distance

R

from the axis of rotation. Add their contributions to the moment of inertia using the formula

I = m R^{2}

, where

m

is the mass of each block.

Step 4: Apply the conservation of angular momentum. Set the initial angular momentum of the turntable equal to the final angular momentum of the system (turntable + blocks). Use the formula

L = I ω

, where

ω

is the angular velocity. Solve for the final angular velocity

ω

Step 5: Convert the final angular velocity from radians per second back to revolutions per minute (rpm) using the conversion factor

\frac{60}{2π}

. This will give the turntable's angular velocity in rpm after the blocks stick.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Conservation of Angular Momentum

The principle of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of that system remains constant. In this scenario, the initial angular momentum of the turntable must equal the final angular momentum after the blocks stick to it, allowing us to calculate the new angular velocity.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotation and depends on the mass distribution relative to the axis of rotation. For the turntable and the added blocks, we need to calculate the total moment of inertia before and after the blocks land to determine the new angular velocity.

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object around an axis, typically measured in radians per second or revolutions per minute (rpm). In this problem, we need to find the new angular velocity of the turntable after the blocks have been added, which will involve using the conservation of angular momentum and the calculated moment of inertia.