The angular velocity of a rotating disc decreases linearly with angular displacement from 60 rev/min to zero during 10 rev . Determine the angular velocity of the disc 3 sec after it begins to slow down

Options

A. `(20pi)/10` rad/sec
B. `(17pi)/10` rad/sec
C. `(7pi)3` rad/sec
D. `(10pi)/3` rad/sec

Answer

To solve the problem, we need to determine the angular velocity of a rotating disc that decreases linearly from 60 revolutions per minute (rev/min) to 0 over a displacement of 10 revolutions. We will find the angular velocity after 3 seconds of deceleration.

### Step-by-Step Solution:

1. **Convert Initial Angular Velocity to Radians per Second**:
- The initial angular velocity (ω_initial) is given as 60 rev/min.
- To convert this to radians per second, we use the conversion factor \(2\pi\) radians per revolution and \(60\) seconds per minute:
\[ \omega_{\text{initial}} = 60 \, \text{rev/min} \times \frac{2\pi \, \text{radians}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} = 2\pi \, \text{radians/s} \]

2. **Determine Angular Displacement in Radians**:
- The angular displacement (θ) is given as 10 revolutions.
- Converting this to radians:
\[ \theta = 10 \, \text{rev} \times 2\pi \, \text{radians/rev} = 20\pi \, \text{radians} \]

3. **Calculate Angular Acceleration**:
- Since the angular velocity decreases linearly, we can use the equation:
\[ \omega_{\text{final}}^2 = \omega_{\text{initial}}^2 + 2\alpha\theta \]
- Here, \(\omega_{\text{final}} = 0\) (the disc stops), \(\omega_{\text{initial}} = 2\pi\), and \(\theta = 20\pi\).
- Plugging in the values:
\[ 0 = (2\pi)^2 + 2\alpha(20\pi) \]
- Simplifying gives:
\[ 0 = 4\pi^2 + 40\pi\alpha \]
- Rearranging for \(\alpha\):
\[ 40\pi\alpha = -4\pi^2 \implies \alpha = -\frac{4\pi^2}{40\pi} = -\frac{\pi}{10} \, \text{radians/s}^2 \]

4. **Find Angular Velocity After 3 Seconds**:
- We use the angular velocity equation:
\[ \omega_{\text{final}} = \omega_{\text{initial}} + \alpha t \]
- Here, \(t = 3 \, \text{s}\), \(\omega_{\text{initial}} = 2\pi\), and \(\alpha = -\frac{\pi}{10}\):
\[ \omega_{\text{final}} = 2\pi + \left(-\frac{\pi}{10}\right)(3) \]
- Simplifying:
\[ \omega_{\text{final}} = 2\pi - \frac{3\pi}{10} = \frac{20\pi}{10} - \frac{3\pi}{10} = \frac{17\pi}{10} \, \text{radians/s} \]

### Final Answer:
The angular velocity of the disc after 3 seconds is \(\frac{17\pi}{10} \, \text{radians/s}\).