/*! 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 66 A car is moving at a rate of 65 ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 66

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 number of revolutions per minute the wheels are rotating is approximately 826.6 revolutions per minute and the angular speed of the wheels is approximately 5193 radians per minute.

Step by step solution

01

Convert speed to feet per minute

The speed of the car is given in miles per hour. We know that 1 mile = 5280 feet and 1 hour = 60 minutes. So, 65 miles per hour can be converted into feet per minute using the formula: \[ Speed_{ft/min} = Speed_{miles/hr} \times \frac{5280 ft}{1 mile} \times \frac{1 hr}{60 min} \]
02

Calculate number of revolutions per minute

The number of revolutions per minute can be determined by dividing the speed (in feet per minute) by the circumference of the wheel (in feet). The circumference of a circle is given by \(\pi \times D\), where D is the diameter of the circle. Applying this to the exercise, we get: \[ Revolutions_{per min} = \frac{Speed_{ft/min}}{\pi \times D_{ft}} \]
03

Calculate the angular speed in radians per minute

Angular speed is the rate at which an object moves through an angle. It is measured in radians per unit time. As earlier stated, \(1 \text{ revolution} = 2\pi \text{ radians}\). Thus, to find the angular speed in radians per minute, we multiply the number of revolutions per minute by \(2\pi\), or: \[Angular Speed_{rad/min} = Revolutions_{per min} \times 2\pi \]

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