A pulley 1 metre in diameter rotating at 600 rpm is brought to rest in 80s by a constant force of frication on its shaft. How many revolutions does it make before coming to rest ?

Options

A. 200
B. 400
C. 300
D. 500

Explanation

To solve the problem of how many revolutions the pulley makes before coming to rest, we can follow these steps:

### Step 1: Convert the initial angular velocity from RPM to radians per second.
The initial angular velocity (ω) in revolutions per minute (RPM) can be converted to radians per second using the formula:

\[ \omega = \text{RPM} \times \frac{2\pi \text{ radians}}{1 \text{ revolution}} \times \frac{1 \text{ minute}}{60 \text{ seconds}} \]

Given that the pulley is rotating at 600 RPM:

\[ \omega = 600 \times \frac{2\pi}{60} = 60\pi \text{ radians/second} \]

### Step 2: Use the formula for angular deceleration.
Since the pulley comes to rest, the final angular velocity (ω_f) is 0. The time (t) taken to come to rest is given as 80 seconds. We can find the angular deceleration (α) using the formula:

\[ \alpha = \frac{\omega_f - \omega}{t} \]

Substituting the values:

\[ \alpha = \frac{0 - 60\pi}{80} = -\frac{60\pi}{80} = -\frac{3\pi}{4} \text{ radians/second}^2 \]

### Step 3: Calculate the total angular displacement (θ) before coming to rest.
We can use the formula for angular displacement when initial angular velocity, final angular velocity, and time are known:

\[ \theta = \omega t + \frac{1}{2} \alpha t^2 \]

Substituting the values we have:

\[ \theta = (60\pi)(80) + \frac{1}{2} \left(-\frac{3\pi}{4}\right)(80^2) \]

Calculating each term:

1. First term:
\[ 60\pi \times 80 = 4800\pi \text{ radians} \]

2. Second term:
\[ \frac{1}{2} \left(-\frac{3\pi}{4}\right)(6400) = -\frac{3\pi}{8} \times 6400 = -2400\pi \text{ radians} \]

Now, adding these two terms:

\[ \theta = 4800\pi - 2400\pi = 2400\pi \text{ radians} \]

### Step 4: Convert angular displacement from radians to revolutions.
To find the number of revolutions (N), we convert radians to revolutions using the formula:

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

Substituting the value of θ:

\[ N = \frac{2400\pi}{2\pi} = 1200 \text{ revolutions} \]

### Final Answer:
The pulley makes **1200 revolutions** before coming to rest.

---