Ch. 10 - Rotational Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 10 - Rotational Motion

Problem 18a

Chapter 10, Problem 18a

The axle of a wheel is mounted on supports that rest on a rotating turntable as shown in Fig. 10–52. The wheel has angular velocity ω₁ = 48.0 rad/s about its axle, and the turntable has angular velocity ω₂ = 35.0 rad/s about a vertical axis. (Note arrows showing these motions in the figure.) What are the directions of $omega sub 1 right arrow$ and $omega sub 2 right arrow$ at the instant shown?

Verified step by step guidance

Understand the problem: The question involves angular velocities of two rotating systems—a wheel and a turntable. The goal is to determine the directions of the angular velocity vectors (ω₁ and ω₂) based on the given information and the right-hand rule.

Recall the right-hand rule for angular velocity: To determine the direction of an angular velocity vector, curl the fingers of your right hand in the direction of the rotation, and your thumb will point in the direction of the angular velocity vector.

Analyze ω₁ (angular velocity of the wheel): The wheel rotates about its axle. Use the right-hand rule to determine the direction of ω₁. If the wheel rotates clockwise or counterclockwise (as indicated in the figure), the direction of ω₁ will be along the axle, either pointing outward or inward.

Analyze ω₂ (angular velocity of the turntable): The turntable rotates about a vertical axis. Use the right-hand rule to determine the direction of ω₂. If the turntable rotates clockwise or counterclockwise (as viewed from above), the direction of ω₂ will be either upward (positive z-axis) or downward (negative z-axis).

Combine the results: Clearly state the directions of ω₁ and ω₂ based on the analysis above. Ensure that the directions are consistent with the figure and the given rotational motions.

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Key Concepts

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

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object around an axis. It is defined by both its magnitude, which indicates how fast the object is rotating, and its direction, which follows the right-hand rule. In this context, the angular velocities ω₁ and ω₂ describe the rotations of the wheel and the turntable, respectively.

Right-Hand Rule

The right-hand rule is a mnemonic used to determine the direction of angular quantities such as angular velocity and torque. By curling the fingers of the right hand in the direction of rotation, the thumb points in the direction of the angular velocity vector. This rule is essential for visualizing the orientation of the angular velocities ω₁ and ω₂ in the given problem.

Relative Motion

Relative motion refers to the analysis of the motion of an object as observed from a particular reference frame. In this scenario, the wheel's rotation (ω₁) is observed from the rotating turntable (ω₂). Understanding relative motion is crucial for determining how the two angular velocities interact and affect the overall motion of the system.

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Intro to Relative Motion (Relative Velocity)

Related Practice

Textbook Question

How fast (in rpm) must a centrifuge rotate if a particle 8.0 cm from the axis of rotation is to experience an acceleration of 100,000 g’s?

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Textbook Question

A turntable of radius R₁ is turned by a circular rubber roller of radius R₂ in contact with it at their outer edges. What is the ratio of their angular velocities, ω₁/ω₂?

Textbook Question

(II) A child rolls a ball on a level floor 3.1 m to another child. If the ball makes 12.0 revolutions, what is its diameter?

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Textbook Question

The axle of a wheel is mounted on supports that rest on a rotating turntable as shown in Fig. 10–52. The wheel has angular velocity ω₁ = 48.0 rad/s about its axle, and the turntable has angular velocity ω₂ = 35.0 rad/s about a vertical axis. (Note arrows showing these motions in the figure? What is the resultant angular velocity of the wheel, as seen by an outside observer, at the instant shown? Give the magnitude and direction.

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Textbook Question

Pilots can be tested for the stresses of flying high-speed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 26 complete revolutions before reaching its final speed. What was its final angular speed in rpm?

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Textbook Question

The axle of a wheel is mounted on supports that rest on a rotating turntable as shown in Fig. 10–52. The wheel has angular velocity ω₁ = 48.0 rad/s about its axle, and the turntable has angular velocity ω₂ = 35.0 rad/s about a vertical axis. (Note arrows showing these motions in the figure.) What is the magnitude and direction of the angular acceleration of the wheel at the instant shown? Take the 𝒵 axis vertically upward and the direction of the axle at the moment shown to be the 𝓍 axis pointing to the right.

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