The angular speed of a point on a planet is 

StartFraction 3 pi Over 8 

radian per hour. The equator lies on a circle of radius approximately 

9000

miles. Find the linear​ velocity, in miles per​ hour, of a point on the equator.




To find the linear velocity of a point on the equator, we can use the relationship between angular speed and linear speed. The formula is:

\[
v = r \cdot \omega
\]

where:
- \(v\) is the linear velocity,
- \(r\) is the radius of the circle (the equator in this case),
- \(\omega\) is the angular speed.

### Given:
- Angular speed, \(\omega = \frac{3\pi}{8}\) radians per hour,
- Radius, \(r = 9000\) miles.

### Step-by-Step Calculation:

1. **Substitute the values into the formula**:
   
   \[
   v = 9000 \times \frac{3\pi}{8} \text{ miles per hour}
   \]

2. **Multiply the terms**:
   
   \[
   v = 9000 \times \frac{3\pi}{8} = \frac{27000\pi}{8} \text{ miles per hour}
   \]

3. **Simplify the expression**:
   
   \[
   v = \frac{27000\pi}{8} = 3375\pi \text{ miles per hour}
   \]

4. **Approximate the value of \(\pi \approx 3.1416\)** to find the linear velocity:

   \[
   v \approx 3375 \times 3.1416 \approx 10602.9 \text{ miles per hour}
   \]

### Final Answer:
The linear velocity of a point on the equator is approximately **10602.9 miles per hour**.

Would you like more details or have any questions?

#### Related Questions:
1. What is the linear velocity in kilometers per hour?
2. How long would it take for a point on the equator to travel one complete revolution around the planet?
3. If the radius of the planet were doubled, what would be the new linear velocity?
4. What is the angular speed in degrees per hour?
5. How would the linear velocity change if the angular speed were doubled?

#### Tip:
To convert between linear and angular velocity, always remember the formula \( v = r \cdot \omega \), where \( r \) is the radius, and \( \omega \) is the angular speed in radians per unit time.

Learn how to calculate the linear velocity of a point on a planet's equator using angular speed and radius. This problem involves applying the linear velocity formula and understanding basic trigonometric principles. Suitable for high school students.

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