Ch 10: Dynamics of Rotational Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 10: Dynamics of Rotational Motion

Problem 45a

Chapter 10, Problem 45a

A large wooden turntable in the shape of a flat uniform disk has a radius of 2.00 m and a total mass of 120 kg. The turntable is initially rotating at 3.00 rad/s about a vertical axis through its center. Suddenly, a 70.0-kg parachutist makes a soft landing on the turntable at a point near the outer edge. Find the angular speed of the turntable after the parachutist lands. (Assume that you can treat the parachutist as a particle.)

Verified step by step guidance

First, understand that this problem involves the conservation of angular momentum. The initial angular momentum of the system (turntable) must equal the final angular momentum of the system (turntable plus parachutist) because no external torques are acting on the system.

Calculate the initial angular momentum of the turntable. The moment of inertia $ I $ of a disk is given by $ I = \frac{1}{2} m r^2 $, where $ m $ is the mass and $ r $ is the radius. Use this formula to find the initial moment of inertia of the turntable.

Multiply the initial moment of inertia by the initial angular speed $ \omega $ to find the initial angular momentum $ L_{initial} = I_{turntable} \times \omega_{initial} $.

Next, calculate the moment of inertia of the parachutist. Since the parachutist can be treated as a particle, use $ I = m r^2 $, where $ m $ is the mass of the parachutist and $ r $ is the distance from the axis of rotation (the radius of the turntable).

Finally, set the initial angular momentum equal to the final angular momentum. The final angular momentum is the sum of the angular momentum of the turntable and the parachutist: $ L_{final} = (I_{turntable} + I_{parachutist}) \times \omega_{final} $. Solve for $ \omega_{final} $ to find the angular speed of the turntable after the parachutist lands.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Conservation of Angular Momentum

The conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of the system remains constant. In this problem, the turntable and parachutist system is isolated, so the initial angular momentum of the turntable must equal the final angular momentum of the combined system after the parachutist lands.

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 a disk, the moment of inertia is calculated as I = (1/2)MR². When the parachutist lands on the turntable, the system's total moment of inertia changes, affecting the angular speed.

Angular Speed

Angular speed is the rate at which an object rotates or revolves around an axis, measured in radians per second. In this scenario, the initial angular speed of the turntable is given, and the task is to find the new angular speed after the parachutist lands, using the conservation of angular momentum and the updated moment of inertia.

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