Question

A plank with a box on it at one end is gradually raised about the other end.As the angle of inclination with the horizontal reaches ${30}^o$ , the box starts to slip and slides $4m$ down the plank in $4s$. The coefficient of static and kinetic friction between the box and the plank will be respectively

Answer 1

At $\theta ={30}^o$

Box is about to slip, hence

$mg\sin \theta ={\mu }_{s}mg\cos \theta$

$\therefore {\mu }_s=\tan \theta$

$=\tan {30}^o=\frac{1}{\sqrt{3}}$

Now, $s=ut+\frac{1}{2}a{t}^2$

$4=0+\frac{1}{2}a\times 4^2$

$\therefore a=\frac{8}{16}=\frac{1}{2}$

But $a=\frac{mg\sin \theta -{\mu }_{k}mg\cos \theta }{m}$

$\frac{1}{2}=g[\sin {30}^o-{\mu }_k\cos {30}^o]$

$\frac{1}{20}=\frac{1}{2}-{\mu }_k\times \frac{\sqrt{3}}{2}$

$\therefore {\mu }_k\times \frac{\sqrt{3}}{2}=\frac{1}{2}-\frac{1}{20}$

$\therefore {\mu }_k=\frac{9}{20}\times \frac{2}{\sqrt{3}}$

$=\frac{3\sqrt{3}}{10}$