Question

A soccer ball, which has a circumference of \(70.0 \mathrm{~cm}\), rolls \(14.0 \mathrm{~m}\) in \(3.35 \mathrm{~s}\). What was the average angular speed of the ball during this time?

Short answer

The average angular speed is approximately 37.5 rad/s.

Step-by-step solution

Determine the Radius

The circumference of a circle is related to its radius by the formula: \[ C = 2\pi r \]where \( C = 70.0 \text{ cm} \). To find the radius \( r \), rearrange the formula:\[ r = \frac{C}{2\pi} = \frac{70.0 \text{ cm}}{2\pi} \approx 11.14 \text{ cm} \].

Convert the Rolling Distance to Revolutions

First, convert the rolling distance to centimeters: \[ 14.0 \text{ m} = 1400 \text{ cm} \]. The number of revolutions \( n \) the ball makes is \[ n = \frac{1400 \text{ cm}}{70.0 \text{ cm/rev}} = 20 \text{ rev} \].

Calculate the Angular Distance

The angular distance in radians can be found using: \[ \theta = n \times 2\pi \text{ radians/rev} = 20 \times 2\pi \approx 125.66 \text{ rad} \].

Determine the Average Angular Speed

Angular speed \( \omega \) is calculated by dividing the angular distance by the time:\[ \omega = \frac{\theta}{t} = \frac{125.66 \text{ rad}}{3.35 \text{ s}} \approx 37.5 \text{ rad/s} \].

Key concepts

Circumference

The circumference is the distance around the outer edge of a circle, much like the perimeter of polygons. You can think of it as the circle's total length if it were unraveled into a straight line. To calculate the circumference, use the formula:
\[ C = 2\pi r \]where \( C \) is the circumference, \( \pi \approx 3.14159 \), and \( r \) is the radius of the circle. For a soccer ball with a circumference of 70 cm, the radius is
- \[ \frac{70.0 \text{ cm}}{2\pi} \approx 11.14 \text{ cm} \]Understanding the circle's circumference helps us assess how far the ball will roll with each revolution.

Revolutions

A revolution refers to one complete turn around the circle's circumference. When the soccer ball rolls, each complete turn is one revolution. To determine how far the ball has traveled in one revolution, realize that it covers a distance equal to its circumference.
A key piece of exercise involves converting a rolling distance into revolutions. Since the ball travels 1400 cm, and its circumference is 70 cm, the ball makes this many revolutions:

• \( n = \frac{1400 \text{ cm}}{70 \text{ cm/rev}} = 20 \text{ rev} \)

This calculation tells us that the ball had to make 20 rounds to cover the 14 meters.

Angular Distance

Angular distance is a bit different from regular distance. It measures how far something rotates or turns in a circular manner, expressed in radians. For each revolution of the soccer ball, it covers an angular distance of \( 2\pi \) radians.
As the ball made 20 revolutions, the total angular distance \( \theta \) is:

• \( \theta = 20 \times 2\pi \approx 125.66 \text{ rad} \)

Angular distance helps us understand how much the ball has rotated around its center during its travel.

Radians

Radians are a unit of angular measure used in many areas of mathematics. Unlike degrees, radians are based on the radius of the circle. One full circle is \( 2\pi \) radians, similar to how it's 360 degrees in degree measure.
When using radians:

• An angle that makes a half circle is \( \pi \) radians (180 degrees).
• An angle that makes a quarter circle is \( \frac{\pi}{2} \) radians (90 degrees).

In this exercise, radians enable precise calculation of the ball's rotation, vital for finding angular speed or understanding motion over curved pathways.