Question

A stone whirled at the end of a rope 30cm makes 10 complete revolutions in 2 seconds. Find :
The angular velocity in radians per second
The linear speed
The distance covered

Answer 1

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.