The turntable T of a record player has a mass of 0.75 kg and a radius of gyration kz=125 mm .

step 1 in this question h z1 equals hz2 so 0 .75 times 0 .125 square multiply 2 equals 0 .75 times 0 .1

The turntable T of a record player has a mass of 0.75 kg...

The turntable $T$ of a record player has a mass of $0.75 \mathrm{~kg}$ and a radius of gyration $k_{z}=125 \mathrm{~mm} .$ It is turning freely at $\omega_{T}=2 \mathrm{rad} / \mathrm{s}$ when a $50-\mathrm{g}$ record (thin disk) falls on it. Determine the final angular velocity of the turntable just after the record stops slipping on the turntable.

Key Concepts

Friction and Transition to Rolling Without Slipping

When the record falls onto the spinning turntable, friction between their surfaces acts to eventually eliminate relative motion. Understanding the role of friction helps explain how the record transitions from slipping on the turntable to rolling together with it, at which point the system’s angular momentum is fully shared.

Radius of Gyration

This parameter is defined as the distance from the rotation axis at which the entire mass of the body could be concentrated without changing its moment of inertia. It simplifies the calculation of the moment of inertia for objects with complex mass distributions, such as a turntable.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion and depends on both the mass of the object and how that mass is distributed relative to the axis of rotation. It plays a critical role in calculating how the angular velocity of a system changes when additional mass is added.

Angular Momentum Conservation

This concept states that in an isolated system with no external torques, the total angular momentum remains constant. When the record falls onto the rotating turntable and eventually spins together without slipping, the loss in rotational speed of the turntable can be determined by equating the initial and final angular momentum of the system.

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