A torque of 50 Nm acting on a wheel at rest rotate it through 200 radians in 5 seconds. The angular acceleration (in rad/sec$^2)$ will be :

Question

Accepted Answer

To determine the angular acceleration of the wheel, we can use the equations of rotational motion. Given that a torque of 50 Nm is acting on the wheel, which rotates it through 200 radians in 5 seconds, we need to find the angular acceleration ($\alpha$).

### Explanation for Option 4:

1. **Using the equation of motion for rotational dynamics:**
   $$
   \theta = \omega_0 t + \frac{1}{2} \alpha t^2
   $$
   where:
   - $\theta$ is the angular displacement (200 radians),
   - $\omega_0$ is the initial angular velocity (0 rad/s, since the wheel is at rest),
   - $t$ is the time (5 seconds),
   - $\alpha$ is the angular acceleration.

Substituting the known values:
   $$
   200 = 0 \times 5 + \frac{1}{2} \alpha \times (5)^2
   $$
   $$
   200 = \frac{1}{2} \alpha \times 25
   $$
   $$
   200 = 12.5 \alpha
   $$
   $$
   \alpha = \frac{200}{12.5} = 16 \, \text{rad/sec}^2
   $$

Thus, the angular acceleration is 16 rad/sec$^2$, which corresponds to Option 4.

### Explanation for why other options are incorrect:

2. **Option 1 (4 rad/sec$^2$):**
   - If $\alpha$ were 4 rad/sec$^2$, substituting back into the equation:
     $$
     \theta = \frac{1}{2} \times 4 \times 25 = 50 \, \text{radians}
     $$
   - This does not match the given angular displacement of 200 radians, hence Option 1 is incorrect.

3. **Option 2 (8 rad/sec$^2$):**
   - If $\alpha$ were 8 rad/sec$^2$, substituting back into the equation:
     $$
     \theta = \frac{1}{2} \times 8 \times 25 = 100 \, \text{radians}
     $$
   - This also does not match the given angular displacement of 200 radians, hence Option 2 is incorrect.

4. **Option 3 (12 rad/sec$^2$):**
   - If $\alpha$ were 12 rad/sec$^2$, substituting back into the equation:
     $$
     \theta = \frac{1}{2} \times 12 \times 25 = 150 \, \text{radians}
     $$
   - This still does not match the given angular displacement of 200 radians, hence Option 3 is incorrect.

In conclusion, the calculations clearly show that the correct angular acceleration is 16 rad/sec$^2$, which is why Option 4 is the correct answer.