Sure, let's break down the solution step by step.
### Given:
1. Radius of the circular path: [tex]\( 45 \, \text{cm} \)[/tex]
2. Number of revolutions: 9
3. Time period for the revolutions: [tex]\( 3(5) \)[/tex]
First, let's convert everything into appropriate units and formats.
- Convert the radius into meters:
[tex]\[
\text{Radius} = 45 \, \text{cm} = 0.45 \, \text{m}
\][/tex]
- Convert the time period: [tex]\( 3(5) \)[/tex] is interpreted as [tex]\( 5/3 \)[/tex] seconds.
### Step 1: Calculate the Angular Velocity
The formula for angular velocity ([tex]\( \omega \)[/tex]) is given by:
[tex]\[
\omega = \frac{\text{Number of revolutions} \times 2 \pi}{\text{Time period}}
\][/tex]
Given:
- Number of revolutions ([tex]\( n \)[/tex]) = 9
- Time period ([tex]\( T \)[/tex]) = [tex]\( \frac{5}{3} \, \text{seconds} \)[/tex]
Plugging these values into the formula:
[tex]\[
\omega = \frac{9 \times 2 \pi}{\frac{5}{3}} = \frac{18 \pi}{\frac{5}{3}} = \frac{18 \pi \times 3}{5} = \frac{54 \pi}{5}
\][/tex]
Solving this gives:
[tex]\[
\omega \approx 33.93 \, \text{rad/s}
\][/tex]
### Step 2: Calculate the Linear Velocity
The formula for linear velocity ([tex]\( v \)[/tex]) is given by:
[tex]\[
v = \text{radius} \times \text{angular velocity}
\][/tex]
Given:
- Radius ([tex]\( r \)[/tex]) = 0.45 m
- Angular velocity ([tex]\( \omega \)[/tex]) [tex]\(\approx 33.93 \, \text{rad/s}\)[/tex]
Plugging these values into the formula:
[tex]\[
v = 0.45 \times 33.93
\][/tex]
Solving this gives:
[tex]\[
v \approx 15.27 \, \text{m/s}
\][/tex]
### Conclusion:
- The angular velocity ([tex]\( \omega \)[/tex]) of the stone is approximately [tex]\( 33.93 \, \text{rad/s} \)[/tex].
- The linear velocity ([tex]\( v \)[/tex]) of the stone is approximately [tex]\( 15.27 \, \text{m/s} \)[/tex].
These values indicate how fast the stone is spinning around the circular path and the speed of the stone along the path, respectively.
