There are two inclined surfaces of equal length L and the same angle of inclination 45^ with the horizontal. One is rough and the…

Question

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

1. A. 0.25
2. B. 0.40
3. C. 0.5
4. D. 0.75 (Correct answer)

Correct Answer

Option D — 0.75

Detailed Solution & Explanation

The correct answer is 0.75.

Key Points

• Smooth incline: $a_{smooth}=g\sin\theta$. Rough incline: $a_{rough}=g\sin\theta-\mu_k g\cos\theta$.
• Since $t\propto \dfrac{1}{\sqrt{a}}$, we have $\dfrac{t_{rough}}{t_{smooth}}=\sqrt{\dfrac{a_{smooth}}{a_{rough}}}=2$.
• So $\dfrac{\sin\theta}{\sin\theta-\mu_k\cos\theta}=4$. At $\theta=45^\circ$ ($\sin\theta=\cos\theta$): $\dfrac{1}{1-\mu_k}=4\Rightarrow \mu_k=0.75$.