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Ch 12: Rotation of a Rigid Body
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 12: Rotation of a Rigid BodyProblem 48b
Chapter 12, Problem 48b

A toy gyroscope has a ring of mass M and radius R attached to the axle by lightweight spokes. The end of the axle is distance R from the center of the ring. The gyroscope is spun at angular velocity ω, then the end of the axle is placed on a support that allows the gyroscope to precess. A 120 g, 8.0-cm-diameter gyroscope is spun at 1000 rpm and allowed to precess. What is the precession period?

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Convert the given values into SI units. The mass of the gyroscope is given as 120 g, which should be converted to kilograms: \( M = 0.120 \; \text{kg} \). The diameter is 8.0 cm, so the radius is \( R = 0.08 / 2 = 0.04 \; \text{m} \). The angular velocity is given as 1000 rpm, which should be converted to radians per second: \( \omega = \frac{1000 \times 2\pi}{60} \; \text{rad/s} \).
Calculate the moment of inertia of the gyroscope. Since the gyroscope is modeled as a ring, its moment of inertia about its axis of rotation is given by \( I = M R^2 \). Substitute the values of \( M \) and \( R \) to find \( I \).
Determine the torque due to gravity. The torque is given by \( \tau = M g R \), where \( g \) is the acceleration due to gravity (\( 9.8 \; \text{m/s}^2 \)). Substitute the values of \( M \), \( g \), and \( R \) to calculate \( \tau \).
Relate the precession angular velocity \( \Omega \) to the torque and angular momentum. The precession angular velocity is given by \( \Omega = \frac{\tau}{L} \), where \( L \) is the angular momentum of the gyroscope. The angular momentum is \( L = I \omega \). Substitute \( \tau \), \( I \), and \( \omega \) to find \( \Omega \).
Calculate the precession period. The precession period \( T \) is the time it takes for one complete precession, which is related to the precession angular velocity by \( T = \frac{2\pi}{\Omega} \). Substitute the value of \( \Omega \) to find \( T \).

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Key Concepts

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

Angular Momentum

Angular momentum is a measure of the rotational motion of an object and is defined as the product of the moment of inertia and the angular velocity. For a gyroscope, it plays a crucial role in understanding how the system behaves when subjected to external torques. The conservation of angular momentum is key to analyzing the precession of the gyroscope, as it remains constant unless acted upon by an external force.
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Precession

Precession is the phenomenon where the axis of a spinning object, like a gyroscope, moves in a circular path due to an external torque. This occurs when the gravitational force acts on the center of mass of the gyroscope, causing it to tilt and rotate around a vertical axis. The rate of precession is influenced by the angular momentum of the gyroscope and the torque applied, which is essential for calculating the precession period.

Moment of Inertia

The moment of inertia is a scalar value that quantifies how mass is distributed relative to an axis of rotation. It determines how much torque is needed for a desired angular acceleration. In the context of the gyroscope, the moment of inertia affects both the angular momentum and the precession behavior, as a larger moment of inertia results in slower changes to the gyroscope's orientation when subjected to external forces.
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Related Practice
Textbook Question

A 75 g, 6.0-cm-diameter solid spherical top is spun at 1200 rpm on an axle that extends 1.0 cm past the edge of the sphere. The tip of the axle is placed on a support. What is the top's precession frequency in rpm?

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Textbook Question

A small 300 g ball and a small 600 g ball are connected by a 40-cm-long, 200 g rigid rod. b. What is the rotational kinetic energy if the structure rotates about its center of mass at 100 rpm?

Textbook Question

What is the angular momentum vector of the 500 g rotating bar in FIGURE EX12.44? Give your answer using unit vectors.

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Textbook Question

Determine the moment of inertia about the axis of the object shown in FIGURE P12.52.

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Textbook Question

A small 300 g ball and a small 600 g ball are connected by a 40-cm-long, 200 g rigid rod. a. How far is the center of mass from the 600 g ball?

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Textbook Question

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?

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