/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 74 A figure skater is spinning at a... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 74

A figure skater is spinning at a rate of 1.0 rev/s with her arms outstretched. She then draws her arms in to her chest, reducing her rotational inertia to \(67 \%\) of its original value. What is her new rate of rotation?

Short Answer

Expert verified
Answer: The new rate of rotation is approximately 1.49 rev/s.

Step by step solution

01

Write the conservation of angular momentum equation

Before and after the skater draws her arms in, the conservation of angular momentum equation states that: \( I_i \omega_i = I_f \omega_f \), where \(I_i\) is the initial moment of inertia, \(\omega_i\) is the initial angular velocity, \(I_f\) is the final moment of inertia, and \(\omega_f\) is the final angular velocity.
02

Find the final moment of inertia

According to the problem, the final moment of inertia (\(I_f\)) is 67% of the initial moment of inertia (\(I_i\)). Therefore, \(I_f = 0.67 I_i\)
03

Substitute the known values into the conservation of angular momentum equation

We know the initial angular velocity (\(\omega_i\)) is 1.0 rev/s, so we can rewrite the equation as: \(I_i(1\,\text{rev/s}) = (0.67 I_i) \omega_f\)
04

Solve for the final angular velocity (\(\omega_f\))

We can now solve for the final angular velocity (\(\omega_f\)) by dividing both sides of the equation by \(0.67I_i\): \(\omega_f = \dfrac{I_i(1\,\text{rev/s})}{0.67I_i}\)
05

Simplify and calculate the final angular velocity

Therefore, the final angular velocity is: \(\omega_f = \dfrac{1\,\text{rev/s}}{0.67} \approx 1.49\,\text{rev/s}\) So, when the figure skater draws her arms in to her chest, her new rate of rotation is approximately 1.49 rev/s.

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Most popular questions from this chapter

An experimental flywheel, used to store energy and replace an automobile engine, is a solid disk of mass \(200.0 \mathrm{kg}\) and radius $0.40 \mathrm{m} .\( (a) What is its rotational inertia? (b) When driving at \)22.4 \mathrm{m} / \mathrm{s}(50 \mathrm{mph}),$ the fully energized flywheel is rotating at an angular speed of \(3160 \mathrm{rad} / \mathrm{s} .\) What is the initial rotational kinetic energy of the flywheel? (c) If the total mass of the car is \(1000.0 \mathrm{kg},\) find the ratio of the initial rotational kinetic energy of the flywheel to the translational kinetic energy of the car. (d) If the force of air resistance on the car is \(670.0 \mathrm{N},\) how far can the car travel at a speed of $22.4 \mathrm{m} / \mathrm{s}(50 \mathrm{mph})$ with the initial stored energy? Ignore losses of mechanical energy due to means other than air resistance.

A uniform disk with a mass of \(800 \mathrm{g}\) and radius \(17.0 \mathrm{cm}\) is rotating on friction less bearings with an angular speed of \(18.0 \mathrm{Hz}\) when Jill drops a 120 -g clod of clay on a point \(8.00 \mathrm{cm}\) from the center of the disk, where it sticks. What is the new angular speed of the disk?
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A solid sphere is released from rest and allowed to roll down a board that has one end resting on the floor and is tilted at \(30^{\circ}\) with respect to the horizontal. If the sphere is released from a height of \(60 \mathrm{cm}\) above the floor, what is the sphere's speed when it reaches the lowest end of the board?
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