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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 solution involves converting the distance covered by the tire into the angle of rotation in radians and then finally into the number of revolutions.

Step by step solution

01

Calculate the Radius and Circumference

Calculate the radius of the tire using the diameter \(d = 2.0 ft\). Since \(d=2r\), solve for \(r\) as \(r = d 梅 2\). Then, calculate the tire's circumference using the formula \(C= 2\pi r\).
02

Find Total Distance The Car Will Travel

Since 1 rotation of the tire moves the car the distance of the tire's circumference, multiply the circumference by 60000 miles to get the total distance the car will travel during the warranty period.
03

Convert Distance to Radians

To convert the distance to radians, divide the total distance by the radius.Thus, the angle in radians can be determined.
04

Calculate the Number of Revolutions

Since 1 revolution is equal to \(2\pi\) radians, to figure out how many revolutions the tire has made means dividing the total angle you got in step 3 by \(2\pi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Displacement
In rotational motion, angular displacement measures how much an object has rotated around a fixed point or axis. It is given in terms of the angle, and it helps us understand how far something has turned.
Angular displacement is measured in radians, which represent the ratio of the arc length traveled by an object in circular motion to its radius.
Just imagine spinning a wheel; the amount it turns from its starting position is its angular displacement.
In this exercise, when calculating angular displacement, we consider the entire path a car tire travels during the warranty period.
Radians
Radians are a way to measure angles based on the radius of a circle. One radian occurs when the arc length is equal to the radius.
This makes the concept of radians more natural in mathematics and physics involving circles because it directly relates angles to the geometry of the circle.
Unlike degrees, which split a circle into 360 parts, radians link angles to properties of the circle, such as its radius and circumference.
  • 1 radian = 180/蟺 degrees
  • The full circle is equivalent to 2蟺 radians
This connection helps streamline calculations in rotation, like finding the angle a car tire turns over thousands of miles.
Circumference of a Circle
The circumference of a circle is the total distance around the circle. It's the circular equivalent of a geometric perimeter.
The formula for calculating the circumference is simple but powerful:
\[ C = 2\pi r \]
Here, \( C \) stands for circumference and \( r \) represents the radius of the circle. In the case of a car tire, the circumference tells us how far the car travels for each complete rotation of the tire.
  • If you know the diameter, you can also use \( C = \pi d \)
  • Total distance traveled = (number of rotations) x (circumference)
Understanding the circumference helps in relating the tire's rotation to the distance covered.
Revolutions
Revolutions are a measure of how many complete turns an object makes around a central point or axis. It's a convenient way to quantify rotation in practical situations, like a spinning tire.
Each complete revolution covers a circular path equal to the circumference of the tire.
Since one revolution corresponds to \(2\pi\) radians, converting angular displacement in radians to revolutions is straightforward:
  • Number of revolutions = total angle in radians /\(2\pi\)
  • In practical terms, the higher the number of revolutions, the more the tire has turned.
This concept helps us understand motion in mechanics and is essential for analyzing rotational systems like vehicles.

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Most popular questions from this chapter

A \(600-\mathrm{kg}\) satellite is in a circular orbit about Earth at a height above Earth equal to Earth's mean radius. Find (a) the satellite's orbital speed, (b) the period of its revolution, and (c) the gravitational force acting on it.

It has been suggested that rotating cylinders about \(10 \mathrm{mi}\) long and \(5.0 \mathrm{mi}\) in diameter be placed in space and used as colonies. What angular speed must such a cylinder have so that the centripetal acceleration at its surface equals the free-fall acceleration on Earth?

An athlete swings a \(5.00\) -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?

In Robert Heinlein's The Moon Is a Harsh Mistress, the colonial inhabitants of the Moon threaten to launch rocks down onto Earth if they are not given independence (or at least representation). Assuming a gun could launch a rock of mass \(m\) at twice the lunar escape speed, calculate the speed of the rock as it enters Earth's atmosphere.

ecp A potter's wheel moves uniformly from rest to an angular speed of \(1.00 \mathrm{rev} / \mathrm{s}\) in \(30.0 \mathrm{~s}\). (a) Find its angular acceleration in radians per second per second. (b) Would doubling the angular acceleration during the given period have doubled final angular speed?
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