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Chapter 4: Problem 115

A car is moving at a rate of 65 miles per hour, and the diameter of its wheels is 2 feet. (a) Find the number of revolutions per minute the wheels are rotating. (b) Find the angular speed of the wheels in radians per minute.

Short Answer

Expert verified
The wheels are rotating at approximately \(\frac{5720}{2\pi} revolutions/minute and the angular speed of the wheels is 5720 radians/minute.\)

Step by step solution

01

Calculate the circumference of the wheel

The key to understanding how many times a wheel rotates is to first understand the distance it covers in one rotation, which is its circumference. This can be calculated with the formula \(C = \pi d\), where \(d\) is the diameter of the wheel. Given the wheel diameter is 2 feet, the circumference \(C\) is \(C = \pi * 2ft = 2\pi ft\).
02

Calculate the number of wheel revolutions per minute

The car's speed is given in miles per hour. To find the number of wheel revolutions per minute, first convert this speed into feet per minute. Given 1 mile = 5280 feet and 1 hour = 60 minutes, the car's speed in feet per minute is \(65 miles/hour * 5280 feet/mile / 60 minute/hour = 5720 feet/minute\). To find the number of wheel revolutions in a minute, divide the car's speed in feet per minute by the wheel's circumference. So, \(Number\: of\: Revolutions = \frac{5720 feet/minute}{2\pi feet/revolution} = \frac{5720}{2\pi} revolutions/minute.\)
03

Calculate angular speed in radians per minute

The angular speed is the rate at which an object changes its angle measured in radians in one minute. The relationship between angle in radians and the number of revolutions is that one complete revolution is equivalent to \(2\pi\) radians. To convert the number of wheel revolutions per minute into an angular speed in radians per minute, multiply the number of revolutions by \(2\pi\). This gives \(Angular\: Speed = \frac{5720}{2\pi} revolutions/minute * 2\pi radians/revolution = 5720 radians/minute.\)

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