Question

A given object taken $n$ times more time to slide down a ${45}^o$ rough inclined plane as it takes to slide down a perfectly smooth ${45}^o$ incline. The coefficient of kinetic friction between the object and the incline is :

• A. $\sqrt{1-\frac{1}{n^2}}$
• B. $1-\frac{1}{n^2}$
• C. $\frac{1}{2-n^2}$
• D. $\sqrt{\frac{1}{1-n^2}}$

Answer 1

Answer: (B)

$1-\frac{1}{n^2}$

For a body moving with constant acceleration, the kinematics equation is $s=ut+\frac{1}{2}a{t}^2$

If the initial speed is zero, then the time taken to reach a distance $s$ is $t=\sqrt{\frac{2s}{a}}$

i.e., $t\propto a^{-0.5}$

In the case of a smooth inclined plane, $a_1=g\sin \theta$

In the case of rough inclined plane, $a_2=g(\sin \theta -\mu \cos \theta )$

Time taken to travel down the smooth inclined plane is $t_1=t$

Time taken to travel down the smooth inclined plane is $t_2=nt$

$\frac{t_1}{t_2}=\sqrt{\frac{a_2}{a_1}}\Rightarrow \frac{t_1^2}{t_2^2}=\frac{a_2}{a_1}\Rightarrow \frac{1}{n^2}=\frac{\sin \theta -\mu \cos \theta }{\sin \theta }\Rightarrow \mu =\tan \theta (1-\frac{1}{n^2})$