A wheel with radius 10 inches covers a distance of 40pi inches in one second. What is the angular speed of this wheel?

To find the angular speed of the wheel, we can use the relationship between linear speed and angular speed. The formula for the distance covered by a wheel is:

\[
\text{Distance} = \text{Radius} \times \text{Angular Speed}
\]

However, the angular speed \(\omega\) is typically expressed in radians per second, and the linear speed \(v\) (distance per time) is given by:

\[
v = r \cdot \omega
\]

Where:
- \(v\) is the linear speed,
- \(r\) is the radius of the wheel,
- \(\omega\) is the angular speed.

The linear speed \(v\) can also be calculated by dividing the distance covered by the time. In this case, the wheel covers a distance of \(40\pi\) inches in 1 second. Therefore, the linear speed \(v\) is:

\[
v = 40\pi \text{ inches/second}
\]

Given the radius \(r = 10\) inches, we can find the angular speed \(\omega\) using the formula:

\[
\omega = \frac{v}{r} = \frac{40\pi}{10} = 4\pi \text{ radians/second}
\]

So, the angular speed of the wheel is \(4\pi\) radians per second.

Would you like more details or have any questions?

Here are some related questions:
1. What is the definition of angular speed?
2. How would the angular speed change if the radius of the wheel doubled?
3. How do you convert angular speed from radians per second to revolutions per minute (RPM)?
4. What is the linear speed if the angular speed is \(6\pi\) radians per second for the same wheel?
5. How does the relationship between linear and angular speed apply to gears of different sizes?

**Tip:** Angular speed can be converted to revolutions per minute (RPM) by using the relation \(1 \text{ revolution} = 2\pi \text{ radians}\).

Learn how to calculate the angular speed of a wheel with a radius of 10 inches, covering a distance of 40π inches in one second. Understand the relationship between angular speed, linear speed, and radius with detailed step-by-step explanation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation