Chapter 13, Problem 21SGR

To determine

To find:The angle $\theta$ in radians through which the tire rotates in one second.

Answer to Problem 21SGR

The tire rotates an angle of $\frac{4\pi }{15}$ radians in one second.

Explanation of Solution

Given information:

A bicycle tire makes $8$ revolutions in one minute.

Radius of tire $=r=15$ inches.

We know

One revolution makes $2\pi$ radians

  $1$ revolution $\to 2\pi$ radians

Multiplying with $8$ on both sides

We get

  $\Rightarrow$ $1\times 8$ revolution $\to 2\pi \times 8$ radians

  $\Rightarrow$ $8$ revolutions $\to 16\pi$ radians

Given that

A bicycle tire makes $8$ revolutions in one minute.

It means in

  $60$ seconds $\to 16\pi$ radians

Dividing with $60$ on both sides

We get

  $\Rightarrow$ $\frac{60}{60}$ seconds $\to \frac{16\pi }{60}$ radians

  $\Rightarrow$ $1$ second $\to \frac{4\pi }{15}$ radians

Therefore,

The tire rotates an angle of $\frac{4\pi }{15}$ radians in one second.