Question

An object is placed on the surface of a smooth inclined plane of inclination $\theta$ . It takes time t to reach the bottom. If the same objective is allowed to slide down a rough inclined plane of same inclination , it takes nt to reach the bottom where n is a number greater than 1. The coefficient of friction is given by

• A. = tan $\theta$ $(1-1/n^2)$
• B. = cot $\theta$ $(1-1/n^2)$
• C. = tan $\theta$ $(1-1/n^2{)}^{1/2}$
• D. = cot $\theta$ $(1-1/n^2{)}^{1/2}$

Answer 1

The correct option is A = tan $\theta$ $(1-1/n^2)$

For smooth surface, and initial velocity being 0, the distance travelled
$s=\frac{1}{2}a{t}^2$
$⟹\frac{s}{t^2}=\frac{1}{2}gsin\theta$
...(equation 1)
For rough surface, it is given that the same distance is travelled in nt, hence
$s=\frac{1}{2}(gsin\theta -\mu gcos\theta )n^2{t}^2⟹\frac{s}{t^2}=\frac{1}{2}{n}^2(gsin\theta -\mu gcos\theta )$
...(equation 2)
solving equation 1 and 2, we get
$\mu =(1-\frac{1}{n^2})tan\theta$