A disc of mass 25kg and diameter 0.4m is rotating about its axis at 240rev/min. The tangential force needed to stop it in 20 sec would be -

Options

A. `pi`
B. `2 pi`
C. `0.25 pi`
D. `0.5 pi`

Answer

To solve the problem of finding the tangential force needed to stop a rotating disc, we will follow these steps:

### Step 1: Calculate the Moment of Inertia (I) of the Disc
The moment of inertia \(I\) for a disc rotating about its axis is given by the formula:

\[ I = \frac{1}{2} m r^2 \]

Where:
- \(m = 25 \, \text{kg}\) (mass of the disc)
- The radius \(r\) can be calculated from the diameter. Given the diameter is \(0.4 \, \text{m}\), the radius \(r\) is:

\[ r = \frac{0.4}{2} = 0.2 \, \text{m} \]

Now substituting the values into the moment of inertia formula:

\[ I = \frac{1}{2} \times 25 \, \text{kg} \times (0.2 \, \text{m})^2 \]

Calculating this:

\[ I = \frac{1}{2} \times 25 \times 0.04 = \frac{1}{2} \times 1 = 0.5 \, \text{kg m}^2 \]

### Step 2: Convert the Angular Velocity (ω) from rev/min to rad/s
The angular velocity in revolutions per minute (rev/min) needs to be converted to radians per second (rad/s):

\[ \omega = 240 \, \text{rev/min} \times \frac{2\pi \, \text{rad}}{1 \, \text{rev}} \times \frac{1 \, \text{min}}{60 \, \text{s}} \]

Calculating this:

\[ \omega = 240 \times \frac{2\pi}{60} = 240 \times \frac{\pi}{30} = 8\pi \, \text{rad/s} \]

### Step 3: Calculate the Angular Deceleration (α)
To stop the disc in 20 seconds, we need to find the angular deceleration \(α\). The final angular velocity \(ω_f\) is \(0 \, \text{rad/s}\) (since it stops), and the initial angular velocity \(ω_i\) is \(8\pi \, \text{rad/s}\). Using the formula:

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

Substituting the values:

\[ \alpha = \frac{0 - 8\pi}{20} = -\frac{8\pi}{20} = -\frac{2\pi}{5} \, \text{rad/s}^2 \]

### Step 4: Calculate the Torque (τ) Required to Stop the Disc
The torque \(τ\) required to stop the disc can be calculated using the formula:

\[ \tau = I \alpha \]

Substituting the values of \(I\) and \(α\):

\[ \tau = 0.5 \, \text{kg m}^2 \times -\frac{2\pi}{5} \, \text{rad/s}^2 \]

Calculating this:

\[ \tau = -\frac{1\pi}{5} \, \text{N m} \]

### Step 5: Calculate the Tangential Force (F)
The tangential force \(F\) can be calculated using the relationship between torque and force:

\[ \tau = F \cdot r \]

Rearranging for \(F\):

\[ F = \frac{\tau}{r} \]

Substituting the values:

\[ F = \frac{-\frac{1\pi}{5}}{0.2} \]

Calculating this:

\[ F = -\frac{1\pi}{5 \times 0.2} = -\frac{1\pi}{1} = -\pi \, \text{N} \]

### Final Answer
The tangential force needed to stop the disc in 20 seconds is:

\[ F = -\pi \, \text{N} \approx -3.14 \, \text{N} \]