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Chapter 4: Problem 64

A carousel with a 50 -foot diameter makes 4 revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed (in feet per minute) of the platform rim of the carousel.

Short Answer

Expert verified
The angular speed of the carousel is \(8\pi\) radians per minute and the linear speed of the platform rim of the carousel is \(200\pi\) feet per minute.

Step by step solution

01

Convert revolutions to radians

The carousel makes 4 revolutions per minute. To convert this to radians, use the fact that one revolution equals \(2\pi\) radians. Multiply the number of revolutions by \(2\pi\). So, 4 revolutions would be \(4 \times 2\pi = 8\pi\) radians.
02

Calculate the angular speed

The above result is the angular speed of the carousel in radians per minute. Thus, the angular speed is \(8\pi\) radians per minute.
03

Calculate the radius

The radius of the carousel is half the diameter. The diameter is given as 50 feet, so the radius is \( \frac{50}{2} = 25 \) feet.
04

Calculate the linear speed

The linear speed of the rim is the product of the radius and the angular speed. Calculate this as \( 25 \times 8\pi = 200\pi \) feet per minute.

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