To solve the problem, let's break it down step by step.
1. Finding the Angular Velocity in Radians per Second
The angular velocity (ω) in radians per second can be calculated using the formula:
\[ \omega = \frac{\text{Total radians}}{\text{Time (seconds)}} \]
First, we need to find the total radians for 10 complete revolutions. One complete revolution is \(2\pi\) radians. So, for 10 revolutions:
\[ \text{Total radians} = 10 \times 2\pi = 20\pi \text{ radians} \]
Now, we substitute this in our formula:
\[ \omega = \frac{20\pi \text{ radians}}{2 \text{ seconds}} = 10\pi \text{ radians/second} \]
2. Finding the Linear Speed
The linear speed (v) can be calculated using the relationship between linear speed and angular velocity:
\[ v = r \cdot \omega \]
where \(r\) is the radius (length of the rope) and \(\omega\) is the angular velocity.
Given that the radius \(r = 30 , \text{cm} = 0.3 , \text{m}\), we substitute the values we have:
\[ v = 0.3 , \text{m} \cdot 10\pi , \text{radians/second} \]
Calculating this gives:
\[ v = 3\pi , \text{m/s} \approx 9.42 , \text{m/s} , (\text{using } \pi \approx 3.14) \]
3. Finding the Distance Covered
The distance covered (d) in linear motion can be calculated using the formula for linear distance:
\[ d = v \cdot t \]
where \(t\) is the time in seconds. We have:
\[ d = 9.42 , \text{m/s} \cdot 2 , \text{s} = 18.84 , \text{m} \]
Summary of Results
- Angular Velocity: \(10\pi , \text{radians/second} \approx 31.42 , \text{radians/second}\)
- Linear Speed: \(3\pi , \text{m/s} \approx 9.42 , \text{m/s}\)
- Distance Covered: \(18.84 , \text{m}\)
These are the results for the given problem.