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 18c

Chapter 10, Problem 18c

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.

Verified step by step guidance

Understand the problem: The wheel has two angular velocities, one about its own axle (ω₁) and another due to the rotation of the turntable (ω₂). We are tasked with finding the angular acceleration of the wheel, which involves determining the rate of change of the angular velocity vector.

Break down the angular velocity vectors: Represent ω₁ and ω₂ in terms of their components. ω₁ is along the 𝓍-axis (to the right), so ω₁ = (48.0 rad/s)𝓍̂. ω₂ is about the vertical 𝒵-axis, so ω₂ = (35.0 rad/s)𝒵̂.

Determine the angular velocity vector of the wheel: The total angular velocity of the wheel is the vector sum of ω₁ and ω₂. Write this as ω = ω₁𝓍̂ + ω₂𝒵̂ = (48.0 rad/s)𝓍̂ + (35.0 rad/s)𝒵̂.

Find the angular acceleration: The angular acceleration is the time derivative of the angular velocity vector, α = dω/dt. Since ω₁ is constant, its derivative is zero. However, ω₂ causes the direction of ω₁ to change as the turntable rotates. This results in a change in the angular velocity vector, which contributes to α.

Calculate the magnitude and direction of α: Use the cross product of ω₂ and ω₁ to find the angular acceleration vector, α = ω₂ × ω₁. The magnitude of α is |α| = |ω₂||ω₁|sin(θ), where θ = 90° (since ω₁ and ω₂ are perpendicular). The direction of α is given by the right-hand rule, perpendicular to both ω₁ and ω₂.

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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 measured in radians per second (rad/s) and indicates both the speed of rotation and the direction of the axis of rotation. In this problem, the wheel's angular velocity (ω₁) and the turntable's angular velocity (ω₂) are crucial for determining the overall motion of the system.

Angular Acceleration

Angular acceleration is the rate of change of angular velocity over time, also expressed as a vector. It indicates how quickly an object is speeding up or slowing down its rotation. In this scenario, calculating the angular acceleration of the wheel involves understanding how the wheel's rotation is affected by both its own motion and the motion of the turntable.

Reference Frames

Reference frames are coordinate systems used to measure and describe the motion of objects. In this question, the Z-axis is defined as vertically upward, and the X-axis is aligned with the axle of the wheel. Understanding the reference frame is essential for accurately determining the direction of angular acceleration and how it relates to the angular velocities of both the wheel and the turntable.

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Related Practice

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 are the directions of $omega sub 1 right arrow$ and $omega sub 2 right arrow$ at the instant shown?

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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

The bolts on the cylinder head of an engine require tightening to a torque of 95 m-N. If the six-sided bolt head is 15 mm across (Fig. 10–55), estimate the force applied near each of the six points by a wrench.

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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

The angular acceleration of a wheel, as a function of time, is α = 4.2 t² ― 9.0 t , where α is in rad/s² and t in seconds. If the wheel starts from rest (θ = 0 , ω = 0, at t = 0), determine a formula for the angular position θ, both as a function of time.

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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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