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 44

Chapter 10, Problem 44

A solid wood door 1.00 m wide and 2.00 m high is hinged along one side and has a total mass of 40.0 kg. Initially open and at rest, the door is struck at its center by a handful of sticky mud with mass 0.500 kg, traveling perpendicular to the door at 12.0 m/s just before impact. Find the final angular speed of the door. Does the mud make a significant contribution to the moment of inertia?

Verified step by step guidance

First, calculate the initial angular momentum of the mud before it hits the door. Use the formula for linear momentum, $ p = mv $, where $ m $ is the mass of the mud and $ v $ is its velocity. Since the mud hits the door at its center, the distance $ r $ from the hinge is half the width of the door, 0.5 m. The angular momentum $ L $ is given by $ L = r \times p $.

Next, determine the moment of inertia of the door about the hinge. For a rectangular door, the moment of inertia $ I $ is given by $ I = \frac{1}{3} m L^2 $, where $ m $ is the mass of the door and $ L $ is its width.

Calculate the moment of inertia of the mud. Since the mud sticks to the door at a distance $ r $ from the hinge, its moment of inertia is $ I_{mud} = m_{mud} \times r^2 $, where $ m_{mud} $ is the mass of the mud.

Add the moment of inertia of the door and the mud to get the total moment of inertia $ I_{total} = I_{door} + I_{mud} $.

Finally, use the conservation of angular momentum to find the final angular speed $ \omega $ of the door. The initial angular momentum of the system is equal to the final angular momentum, so $ L_{initial} = I_{total} \times \omega $. Solve for $ \omega $ to find the final angular speed of the door.

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Key Concepts

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

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation. For a rectangular door hinged along one side, it can be calculated using the formula I = (1/3) * m * w^2, where m is the mass and w is the width. This concept is crucial for determining how the door's mass distribution affects its rotational motion.

Conservation of Angular Momentum

The conservation of angular momentum states that if no external torques act on a system, its angular momentum remains constant. In this scenario, the angular momentum before the mud hits the door must equal the angular momentum after the collision, allowing us to solve for the door's final angular speed.

Impact and Collision

When the mud strikes the door, it transfers linear momentum to the door, converting it into angular momentum. Understanding the nature of this collision, where the mud sticks to the door, is essential for calculating the change in the door's rotational motion and assessing the mud's contribution to the system's moment of inertia.

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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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