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 20b

Chapter 10, Problem 20b

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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Determine the total time taken for the centrifuge to complete 26 revolutions. The problem states that this time is 1.0 minute, which can be converted to seconds for consistency in calculations: $ t = 1.0 \text{ min} = 60 \text{ s} $.

Calculate the average angular velocity $ \omega_{\text{avg}} $ during the motion. The formula for average angular velocity is $ \omega_{\text{avg}} = \frac{\Delta \theta}{\Delta t} $, where $ \Delta \theta $ is the total angular displacement in radians and $ \Delta t $ is the time interval. Since there are 26 revolutions, convert this to radians using $ 1 \text{ revolution} = 2\pi \text{ radians} $: $ \Delta \theta = 26 \times 2\pi $.

Substitute $ \Delta \theta $ and $ \Delta t $ into the formula for $ \omega_{\text{avg}} $: $ \omega_{\text{avg}} = \frac{26 \times 2\pi}{60} $. This gives the average angular velocity in radians per second.

Recognize that the centrifuge starts from rest and accelerates uniformly to its final angular speed $ \omega_f $. For uniform angular acceleration, the relationship between the final angular speed, average angular speed, and initial angular speed is $ \omega_{\text{avg}} = \frac{\omega_i + \omega_f}{2} $. Since $ \omega_i = 0 $, simplify to $ \omega_{\text{avg}} = \frac{\omega_f}{2} $.

Solve for the final angular speed $ \omega_f $ in radians per second: $ \omega_f = 2 \cdot \omega_{\text{avg}} $. Finally, convert $ \omega_f $ from radians per second to revolutions per minute (rpm) using the conversion factors $ 1 \text{ revolution} = 2\pi \text{ radians} $ and $ 1 \text{ minute} = 60 \text{ seconds} $: $ \omega_f (\text{rpm}) = \omega_f (\text{rad/s}) \cdot \frac{60}{2\pi} $.

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Key Concepts

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

Angular Speed

Angular speed is a measure of how quickly an object rotates around a central point or axis. It is typically expressed in radians per second or revolutions per minute (rpm). In this context, we need to calculate the final angular speed of the centrifuge after it completes its revolutions.

Revolutions and Time

A revolution refers to a complete turn around a circle. To find the angular speed in rpm, we need to relate the number of revolutions to the time taken. In this case, the centrifuge completes 26 revolutions in 1.0 minute, which allows us to calculate the angular speed by dividing the total revolutions by the time in minutes.

Conversion of Units

Unit conversion is essential in physics to ensure that measurements are in compatible units for calculations. Here, we need to convert the time from minutes to seconds if necessary, but since we are calculating rpm, we can directly use the time in minutes. Understanding how to convert between different units is crucial for accurate results.

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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

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

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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