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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

A uniform diving board, of length \(5.0 \mathrm{m}\) and mass \(55 \mathrm{kg}\) is supported at two points; one support is located \(3.4 \mathrm{m}\) from the end of the board and the second is at \(4.6 \mathrm{m}\) from the end (see Fig. 8.19 ). What are the forces acting on the board due to the two supports when a diver of mass \(65 \mathrm{kg}\) stands at the end of the board over the water? Assume that these forces are vertical. (W) tutorial: plank) [Hint: In this problem, consider using two different torque equations about different rotation axes. This may help you determine the directions of the two forces.]
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