Ch 09: Rotation of Rigid Bodies

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 09: Rotation of Rigid Bodies

Problem 15a

Chapter 9, Problem 15a

A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass 40.0 kg and diameter 75.0 cm. The power is off for 30.0 s, and during this time the flywheel slows due to friction in its axle bearings. During the time the power is off, the flywheel makes 200 complete revolutions. At what rate is the flywheel spinning when the power comes back on?

Verified step by step guidance

Step 1: Convert the initial angular velocity from revolutions per minute (rpm) to radians per second (rad/s). Use the formula \( \omega = \frac{2\pi \times \text{rpm}}{60} \), where \( \omega \) is the angular velocity in rad/s.

Step 2: Calculate the total angular displacement during the power outage. Since the flywheel makes 200 complete revolutions, convert this to radians using \( \theta = 200 \times 2\pi \), where \( \theta \) is the angular displacement in radians.

Step 3: Use the kinematic equation for rotational motion to find the angular acceleration \( \alpha \). The equation is \( \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \), where \( \omega_0 \) is the initial angular velocity, \( t \) is the time, and \( \alpha \) is the angular acceleration. Rearrange to solve for \( \alpha \).

Step 4: Use the angular acceleration \( \alpha \) and the kinematic equation \( \omega = \omega_0 + \alpha t \) to find the final angular velocity \( \omega \) at the end of the 30.0 s period. Here, \( \omega \) is the angular velocity when the power comes back on.

Step 5: Convert the final angular velocity \( \omega \) from radians per second (rad/s) back to revolutions per minute (rpm) using the formula \( \text{rpm} = \frac{\omega \times 60}{2\pi} \). This will give the rate at which the flywheel is spinning when the power comes back on.

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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 measure of how quickly an object rotates around an axis, typically expressed in radians per second or revolutions per minute (rpm). In this scenario, the initial angular velocity of the flywheel is given as 500 rpm, which indicates its speed before the power failure. Understanding angular velocity is crucial for determining how it changes over time due to external factors like friction.

Friction and Deceleration

Friction is a force that opposes the motion of an object, causing it to decelerate. In the case of the flywheel, friction in the axle bearings acts to slow it down when the power is off. The rate of deceleration can be calculated by considering the number of revolutions made during the power outage and the time elapsed, which is essential for finding the final angular velocity when power is restored.

Revolutions and Time Relationship

The relationship between the number of revolutions an object makes and the time taken is fundamental in rotational motion. In this problem, the flywheel makes 200 complete revolutions over 30 seconds while slowing down. This information allows us to calculate the average angular velocity during the power outage, which is necessary to determine the flywheel's speed when the power comes back on.

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Textbook Question

A compact disc (CD) stores music in a coded pattern of tiny pits 10^-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track?

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Textbook Question

CA compact disc (CD) stores music in a coded pattern of tiny pits 10-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line?

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Textbook Question

A bicycle wheel has an initial angular velocity of 1.50 rad/s. (a) If its angular acceleration is constant and equal to 0.200 rad/s², what is its angular velocity at t = 2.50 s? (b) Through what angle has the wheel turned between t = 0 and t = 2.50 s?

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Textbook Question

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. Find the angular acceleration in rev/s² and the number of revolutions made by the motor in the 4.00-s interval.

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Textbook Question

A compact disc (CD) stores music in a coded pattern of tiny pits 10^-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant linear speed of 1.25 m/s. What is the average angular acceleration of a maximum duration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.

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Textbook Question

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

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