A carousel rotates at constant speed and completes a revolution in 10.0 . Find the linear speed of Ariel, who is seated at approximately 2.0m from the center of the carousel.
The Correct Answer and Explanation is :
To find Ariel’s linear speed on the carousel, we need to use the relationship between angular speed and linear speed. Here’s the step-by-step process:
Formula for Linear Speed
The linear speed (v) of an object moving in a circular path is related to its angular speed (\omega) and its distance from the center (radius) (r) by the formula:
[
v = \omega \cdot r
]
Where:
- (v) is the linear speed.
- (\omega) is the angular speed.
- (r) is the radius (distance from the center).
Step 1: Calculate Angular Speed
Angular speed (\omega) is defined as the angle covered per unit time. Since a carousel completes one full revolution in 10 seconds, and a full revolution is (2\pi) radians, we can calculate (\omega) as follows:
[
\omega = \frac{2\pi \, \text{radians}}{T}
]
Where:
- (T = 10.0 \, \text{seconds}) (the time for one revolution).
[
\omega = \frac{2\pi}{10} = 0.628 \, \text{radians/second}
]
Step 2: Calculate Linear Speed
Now that we have the angular speed, we can find Ariel’s linear speed using the formula (v = \omega \cdot r). Here, (r = 2.0 \, \text{meters}).
[
v = 0.628 \, \text{radians/second} \times 2.0 \, \text{meters} = 1.256 \, \text{meters/second}
]
Thus, Ariel’s linear speed is approximately 1.26 meters per second.
Explanation
Linear speed describes how fast an object moves along a path, whereas angular speed describes how fast it rotates around a center point. On a carousel, each rider has the same angular speed, but their linear speeds vary based on their distance from the center. Ariel, sitting 2 meters away, experiences a higher linear speed than someone sitting closer to the center. This is because Ariel covers more ground in the same amount of time, even though both experience the same angular velocity.