Ch. 10 - Rotational Motion

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179당신이 사용하는 게 아니라요?교과서 변경

Giancoli Douglas 5th edition

Ch. 10 - Rotational Motion

문제 20b

10장, 문제 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?

검증된 단계별 안내

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} $.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

관련 실천

교과서 질문

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교과서 질문

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교과서 질문

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교과서 질문

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교과서 질문

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