To solve for the angular velocity and the centripetal acceleration of a body moving around a circle with a given linear velocity and radius, we will follow these steps:
### Given:
- Radius ([tex]\(r\)[/tex]) = 1.0 meters
- Linear velocity ([tex]\(v\)[/tex]) = 6.5 meters per second
### 1. Angular Velocity ([tex]\(\omega\)[/tex])
Angular velocity is the rate of change of the angle traversed by the body per unit time and is given by the formula:
[tex]\[ \omega = \frac{v}{r} \][/tex]
Substitute the given values:
[tex]\[ \omega = \frac{6.5 \, \text{m/s}}{1.0 \, \text{m}} \][/tex]
This simplifies to:
[tex]\[ \omega = 6.5 \, \text{radians per second} \][/tex]
### 2. Centripetal Acceleration ([tex]\(a_c\)[/tex])
Centripetal acceleration is the acceleration that keeps the body moving in a circular path and is directed towards the center of the circle. It is given by the formula:
[tex]\[ a_c = \frac{v^2}{r} \][/tex]
Substitute the given values:
[tex]\[ a_c = \frac{(6.5 \, \text{m/s})^2}{1.0 \, \text{m}} \][/tex]
Calculating the numerator:
[tex]\[ (6.5 \, \text{m/s})^2 = 42.25 \, \text{m}^2/\text{s}^2 \][/tex]
Now, dividing by the radius:
[tex]\[ a_c = \frac{42.25 \, \text{m}^2/\text{s}^2}{1.0 \, \text{m}} \][/tex]
This simplifies to:
[tex]\[ a_c = 42.25 \, \text{m/s}^2 \][/tex]
### Summary:
1. Angular Velocity: [tex]\(\omega = 6.5 \, \text{radians per second}\)[/tex]
2. Centripetal Acceleration: [tex]\(a_c = 42.25 \, \text{m/s}^2\)[/tex]
