If a block moving upon an inclined plane at 30° with a velocity of 5 m/s, stops after 0.5 s. Then coefficient of friction will be nearly :

Question

Accepted Answer

To determine the coefficient of friction for a block moving on an inclined plane, we need to consider the forces acting on the block and use the given information to calculate the frictional force.

### Explanation Supporting Option 2:

1. **Given Data:**
   - Incline angle ($\theta$) = 30°
   - Initial velocity ($u$) = 5 m/s
   - Time ($t$) = 0.5 s
   - Final velocity ($v$) = 0 m/s (since the block stops)

2. **Calculating Acceleration:**
   Using the equation of motion:
   $$
   v = u + at
   $$
   $$
   0 = 5 + a \times 0.5
   $$
   $$
   a = -\frac{5}{0.5} = -10 \, \text{m/s}^2
   $$
   The negative sign indicates deceleration.

3. **Forces on the Inclined Plane:**
   - Gravitational force component along the plane: $mg \sin \theta$
   - Frictional force: $f = \mu mg \cos \theta$

4. **Net Force and Acceleration:**
   The net force causing deceleration is the difference between gravitational force and frictional force:
   $$
   mg \sin \theta - \mu mg \cos \theta = ma
   $$
   $$
   g \sin \theta - \mu g \cos \theta = a
   $$
   Substituting the known values:
   $$
   9.8 \times \frac{1}{2} - \mu \times 9.8 \times \frac{\sqrt{3}}{2} = -10
   $$
   $$
   4.9 - \mu \times 8.49 = -10
   $$
   $$
   \mu \times 8.49 = 4.9 + 10
   $$
   $$
   \mu \times 8.49 = 14.9
   $$
   $$
   \mu = \frac{14.9}{8.49} \approx 0.6
   $$

Thus, the coefficient of friction is approximately 0.6, which corresponds to Option 2.

### Explanation for Incorrect Options:

- **Option 1 (0.5):** This value is too low. The calculated coefficient of friction is approximately 0.6, which is higher than 0.5. Therefore, Option 1 does not provide enough frictional force to stop the block in the given time.

- **Option 3 (0.9):** This value is too high. The calculated coefficient of friction is approximately 0.6, which is significantly lower than 0.9. Therefore, Option 3 overestimates the frictional force required.

- **Option 4 (1.1):** This value is also too high. The calculated coefficient of friction is approximately 0.6, which is much lower than 1.1. Therefore, Option 4 greatly overestimates the frictional force needed.

In conclusion, based on the calculations and logical reasoning, Option 2 (0.6) is the correct answer for the coefficient of friction.