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Chapter 7: Problem 3

The tires on a new compact car have a diameter of \(2.0 \mathrm{ft}\) and are warranted for 60000 miles. (a) Determine the angle (in radians) through which one of these tires will rotate during the warranty period. (b) How many revolutions of the tire are equivalent to your answer in part (a)?

Short Answer

Expert verified
The tire rotates through \(316800000\pi\) radians or makes \(158400000\) revolutions during the warranty period.

Step by step solution

01

Calculate the circumference of the tire

The diameter of the tire is 2 feet. Therefore, its radius is 1 foot. The circumference of the tire can be found using formula for circumference of a circle which is \(2\pi r \). Hence, the circumference of the tire is \(2\pi \times 1 = 2\pi\) feet.
02

Calculate the total rotation angle

The warranty for the tires is 60000 miles. Convert this distance into feet: \(1\) mile \(=5280\) feet, hence \(60000\) miles \(=60000 \times 5280\) feet. The rotation angle in radians is equal to the total distance traveled divided by the radius of the tire. So rotation angle \(=\frac{60000 \times 5280}{1}=316800000 \pi\) radians.
03

Calculate number of revolutions

The rotation angle in number of full revolutions is obtained by dividing the total rotation angle by \(2\pi\). Hence the number of revolutions \(=\frac{316800000\pi}{2\pi}=158400000\) revolutions.

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Most popular questions from this chapter

An athlete swings a \(5.00-\mathrm{kg}\) ball horizontally on the end of a rope. The ball moves in a circle of radius \(0.800 \mathrm{~m}\) at an angular speed of \(0.500 \mathrm{rev} / \mathrm{s}\). What are (a) the tangential speed of the ball and (b) its centripetal acceleration? (c) If the maximum tension the rope can withstand before breaking is \(100 \mathrm{~N}\), what is the maximum tangential speed the ball can have?

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