Question

A smooth block is released at rest on a ${45}^o$ incline and then slides a distance $d$. The time taken to slide is $n$ times as much to slide on rough incline than on a smooth incline. The coefficient of friction is:

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

Answer 1

Answer: (A)

${\mu }_k=1-\frac{1}{n^2}$

On a rough surface

$a=(gsin\theta -\mu gcos\theta )$

$time=nt$

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

$0+\frac{1}{2}[gsin\theta -\mu gcos\theta ](nt{)}^2$ ....(1)

On plane surface

$a=gsin\theta$

time = t

$S=0+\frac{1}{2}\left(gsin\theta \right)t^2$ ...(2)

By solving (1) and (2)

$\mu =1-\frac{1}{n^2}$ [$\mu ={\mu }_k$ has the body is moving]

Hence (A) is corrct answer