Question

There are two inclined surfaces of equal length (L) and same angle of inclination 45° with the horizontal. One of them is rough and the other is perfectly smooth. A given body takes 2 times as much time to slide down on rough surface than on the smooth surface. The coefficient of kinetic friction (µk) between the object and the rough surface is close to

• A. 0.25
• B. 0.40
• C. 0.5
• D. 0.75

Answer 1

Answer: (D)

0.75

Answer 2

Here's how to solve this problem:

1. Acceleration on the smooth surface ($a_s$): $a_s = g \sin\theta = g \sin 45^\circ = \frac{g}{\sqrt{2}}$

2. Acceleration on the rough surface ($a_r$): $a_r = g (\sin\theta - \mu_k \cos\theta) = g \left( \frac{1}{\sqrt{2}} - \mu_k \frac{1}{\sqrt{2}} \right) = \frac{g}{\sqrt{2}} (1 - \mu_k)$

3. Time relation: Since $t = \sqrt{\frac{2L}{a}}$ and $t_r = 2t_s$, we have $a_s = 4a_r$.

4. Solving for $\mu_k$: $\frac{g}{\sqrt{2}} = 4 \left( \frac{g}{\sqrt{2}} (1 - \mu_k) \right)$ $1 = 4 (1 - \mu_k)$ $\mu_k = \frac{3}{4} = 0.75$

Therefore, the coefficient of kinetic friction ($\mu_k$) is 0.75.