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 45b

Chapter 10, Problem 45b

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. Compute the kinetic energy of the system before and after the parachutist lands. Why are these kinetic energies not equal?

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First, calculate the initial kinetic energy of the turntable. The formula for the kinetic energy of a rotating object is $ KE = \frac{1}{2} I \omega^2 $, where $ I $ is the moment of inertia and $ \omega $ is the angular velocity. For a uniform disk, the moment of inertia $ I $ is given by $ I = \frac{1}{2} m r^2 $, where $ m $ is the mass and $ r $ is the radius.

Substitute the values for the turntable into the moment of inertia formula: $ I = \frac{1}{2} \times 120 \text{ kg} \times (2.00 \text{ m})^2 $. Then, use this $ I $ to find the initial kinetic energy using $ \omega = 3.00 \text{ rad/s} $.

Next, calculate the moment of inertia of the system after the parachutist lands. The parachutist can be treated as a point mass at a distance $ r $ from the axis of rotation. The new moment of inertia $ I' $ is the sum of the turntable's moment of inertia and the parachutist's contribution: $ I' = I + m_p r^2 $, where $ m_p $ is the parachutist's mass.

After the parachutist lands, the angular velocity $ \omega' $ changes due to conservation of angular momentum. The initial angular momentum $ L = I \omega $ must equal the final angular momentum $ L' = I' \omega' $. Solve for $ \omega' $ using $ \omega' = \frac{I \omega}{I'} $.

Finally, compute the kinetic energy of the system after the parachutist lands using $ KE' = \frac{1}{2} I' \omega'^2 $. The kinetic energies before and after are not equal because the parachutist landing on the turntable increases the moment of inertia, which decreases the angular velocity, leading to a lower kinetic energy despite the conservation of angular momentum.

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Key Concepts

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

Rotational Kinetic Energy

Rotational kinetic energy is the energy due to the rotation of an object and is given by the formula KE = 0.5 * I * ω², where I is the moment of inertia and ω is the angular velocity. For a uniform disk, the moment of inertia is I = 0.5 * m * r². Understanding this concept is crucial for calculating the kinetic energy of the turntable before and after the parachutist lands.

Conservation of Angular Momentum

The conservation of angular momentum states that if no external torques act on a system, the total angular momentum remains constant. When the parachutist lands on the turntable, the system's angular momentum before and after the landing must be equal, allowing us to find the new angular velocity and subsequently the kinetic energy after the landing.

Inelastic Collision

An inelastic collision is one in which kinetic energy is not conserved, although momentum is. When the parachutist lands on the turntable, the collision is inelastic because the parachutist and turntable move together afterward, leading to a loss of kinetic energy due to the conversion of some energy into other forms, such as heat or sound.

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Related Practice

Textbook Question

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Textbook Question

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Textbook Question

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.)

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