Question

When a box is dropped from a certain height, it reaches the ground with a speed of $v$. If it is released from rest from the same height down a rough inclined plane at an angle of ${45}^{\circ }$ with the horizontal, it reaches the ground with a speed of $v/3$. The coefficient of kinetic friction between the box and the plane is

• A. $\frac{8}{9}$
• B. $\frac{1}{9}$
• C. $\frac{2}{9}$
• D. $\frac{4}{9}$

Answer 1

The correct option is A $\frac{8}{9}$

When dropped from height $h$, speed of the box
$v^2=2gh$ ............(1)
When the box is released along the inclined plane,


Let us assume, work done by gravitational force is $W_g$
Work done by normal force is $W_N$
Work done by frictional force is $W_f$
Applying work energy theorem,
Work done by all forces $=$ Change in kinetic energy
$W_g+W_N+W_f=\Delta K.E$
$\Rightarrow (+mgh)+0+(-\mu mgd)=\frac{1}{2}m(v/3{)}^2-0$
$\Rightarrow mgh-\mu mg\times \frac{h}{\tan {45}^{\circ }}=\frac{m\times 2gh}{18}$ [from (1)]
$\Rightarrow mgh-\mu mgh=\frac{mgh}{9}$
$\Rightarrow \mu =\frac{8}{9}$