Question

Starting from rest, a body slides down a ${45}^o$ inclined plane in twice the time it takes to slide the same distance in the absence of friction. What is the coefficient of friction between the body and the inclined plane?

• A. $\frac{1}{4}$
• B. $\frac{1}{2}$
• C. $\frac{3}{4}$
• D. $\frac{\sqrt{3}}{2}$

Answer 1

The correct option is C $\frac{3}{4}$

Given,

Time taken on rough plane $t_2$ is twice the time taken on smooth plane $t_1$. $(t_2=2t_1)$

Acceleration component on smooth incline plane, $a_s=g\sin 45=\frac{g}{\sqrt{2}}$

Acceleration on rough incline plane, $a_r=g\sin 45-\mu g\cos 45=\frac{g(1-\mu )}{\sqrt{2}}$

If initial velocity of both is zero on incline plane.

Apply kinematic equation

$S=ut+\frac{1}{2}a{t}^2$

$S=\frac{1}{2}a{t}^2$

$t=\sqrt{\frac{2S}{a}}$

Time taken on smooth plane, $t_1=\sqrt{\frac{2S}{g/\sqrt{2}}}$

Time taken on rough plane, $t_2=\sqrt{\frac{2S}{g(1-\mu )/\sqrt{2}}}$

$t_2=2t_1$

$\Rightarrow \sqrt{\frac{2S}{g(1-\mu )/\sqrt{2}}}=2\sqrt{\frac{2S}{g/\sqrt{2}}}$

$\Rightarrow \frac{1}{1-\mu }=4$

$\Rightarrow \mu =\frac{3}{4}=0.75$

Friction coefficient is $\frac{3}{4}$