The speed of a motor decreases from 1200 rpm to 600 rpm in 20s. The total number of rotations it makes before coming to rest is

Options

A. 400
B. 600
C. 800
D. 1000

Answer

To solve the problem, we need to find the total number of rotations a motor makes as its speed decreases from 1200 rpm to 600 rpm over a time period of 20 seconds.

### Step-by-Step Solution:

1. **Convert RPM to Radians per Second:**
- The initial speed (ω₀) is 1200 rpm. To convert this to radians per second:
\[ \omega_0 = 1200 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} = 40\pi \, \text{rad/s} \]
- The final speed (ω) is 600 rpm. Similarly, converting this to radians per second:
\[ \omega = 600 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} = 20\pi \, \text{rad/s} \]

2. **Calculate Angular Acceleration (α):**
- The angular acceleration can be calculated using the formula:
\[ \alpha = \frac{\omega - \omega_0}{t} \]
- Substituting the values:
\[ \alpha = \frac{20\pi - 40\pi}{20} = \frac{-20\pi}{20} = -\pi \, \text{rad/s}^2 \]

3. **Use the Angular Displacement Formula:**
- We can use the equation relating angular displacement (θ), initial angular velocity (ω₀), final angular velocity (ω), and angular acceleration (α):
\[ \omega^2 = \omega_0^2 + 2\alpha\theta \]
- Rearranging gives:
\[ \theta = \frac{\omega^2 - \omega_0^2}{2\alpha} \]

4. **Substituting Known Values:**
- Since the motor comes to rest, the final angular velocity (ω) is 0:
\[ \theta = \frac{0 - (40\pi)^2}{2 \times (-\pi)} \]
- Simplifying:
\[ \theta = \frac{-1600\pi^2}{-2\pi} = \frac{1600\pi^2}{2\pi} = 800\pi \, \text{radians} \]

5. **Convert Angular Displacement to Rotations:**
- To find the number of rotations (n), we convert radians to revolutions:
\[ n = \frac{\theta}{2\pi} = \frac{800\pi}{2\pi} = 400 \]

### Final Answer:
The total number of rotations the motor makes before coming to rest is **400**.

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