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. (a)200
B. (b)300
C. (c)400
D. (d)500

Explanation

To solve the problem step by step, we will follow these calculations:

### Step 1: Convert Angular Velocity to Radians per Second
The initial angular velocity (ω₁) is given as 600 revolutions per minute (rpm). We need to convert this to radians per second.

\[ \omega_1 = 600 \, \text{rpm} \times \frac{2\pi \, \text{radians}}{1 \, \text{revolution}} \times \frac{1 \, \text{minute}}{60 \, \text{seconds}} \]

Calculating this gives:

\[ \omega_1 = 600 \times \frac{2\pi}{60} = 20\pi \, \text{radians/second} \]

### Step 2: Determine Final Angular Velocity
The final angular velocity (ω₂) when the pulley comes to rest is:

\[ \omega_2 = 0 \, \text{radians/second} \]

### Step 3: Calculate Angular Deceleration
The angular deceleration (α) can be calculated using the formula:

\[ \alpha = \frac{\omega_2 - \omega_1}{t} \]

Substituting the known values:

\[ \alpha = \frac{0 - 20\pi}{80} = -\frac{20\pi}{80} = -\frac{\pi}{4} \, \text{radians/second}^2 \]

### Step 4: Calculate Angular Displacement
We can use the following equation of motion for angular displacement (θ):

\[ \omega_2^2 = \omega_1^2 + 2\alpha\theta \]

Rearranging gives:

\[ \theta = \frac{\omega_2^2 - \omega_1^2}{2\alpha} \]

Substituting the values:

\[ \theta = \frac{0^2 - (20\pi)^2}{2 \times -\frac{\pi}{4}} \]

Calculating this gives:

\[ \theta = \frac{-400\pi^2}{-\frac{\pi}{2}} = \frac{400\pi^2 \times 2}{\pi} = 800 \, \text{radians} \]

### Step 5: Calculate the Number of Revolutions
To find the number of revolutions (N), we use the relation:

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

Substituting the value of θ:

\[ N = \frac{800}{2\pi} = \frac{800}{2\pi} = \frac{400}{\pi} \]

### Final Calculation
Since we are interested in the number of revolutions, we can approximate π as 3.14 for practical purposes, but the exact answer in terms of π is:

\[ N \approx 400 \, \text{revolutions} \]

### Conclusion
The pulley makes approximately **400 revolutions** before coming to rest.

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