The motor of an engine is rotating about its axis with an angular velocity of 100 rpm. It comes to rest in 15 s after being switched off. Assuming constant angular deceleration, calculate the number of revolution made by it before coming to rest.

Options

A. `12.5`
B. 40
C. `32.6`
D. `15.6`

Explanation

To solve the problem step by step, we will use the equations of motion for rotational motion.

### Step 1: Convert Angular Velocity from RPM to Radians per Second
The initial angular velocity (ω₀) is given as 100 revolutions per minute (rpm). We need to convert this to radians per second.

\[ \omega_0 = 100 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} \]

Calculating this gives:

\[ \omega_0 = 100 \times \frac{2\pi}{60} = \frac{100 \times 2\pi}{60} = \frac{100\pi}{30} \approx 10.47 \, \text{rad/s} \]

### Step 2: Calculate Angular Deceleration
The motor comes to rest in 15 seconds, which means the final angular velocity (ω) is 0 rad/s. We can use the formula for angular deceleration (α):

\[ \alpha = \frac{\Delta \omega}{\Delta t} = \frac{\omega - \omega_0}{t} \]

Substituting the values:

\[ \alpha = \frac{0 - 10.47}{15} = \frac{-10.47}{15} \approx -0.698 \, \text{rad/s}^2 \]

### Step 3: Calculate the Angular Displacement
We can use the equation for angular displacement (θ) when the initial angular velocity, final angular velocity, and angular acceleration are known:

\[ \theta = \frac{\omega_0^2 - \omega^2}{2\alpha} \]

Substituting the known values:

\[ \theta = \frac{(10.47)^2 - 0^2}{2 \times (-0.698)} \]

Calculating this gives:

\[ \theta = \frac{109.61}{-1.396} \approx -78.5 \, \text{radians} \]

(Note: The negative sign indicates the direction of rotation, but we are interested in the magnitude for the number of revolutions.)

### Step 4: Convert Angular Displacement to Revolutions
To find the number of revolutions (N), we convert the angular displacement from radians to revolutions:

\[ N = \frac{\theta}{2\pi} \]

Substituting the value of θ:

\[ N = \frac{78.5}{2\pi} \approx \frac{78.5}{6.283} \approx 12.5 \, \text{revolutions} \]

### Final Answer
The number of revolutions made by the motor before coming to rest is approximately **12.5 revolutions**.

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