Question

A plank of mass $2\text{kg}$ and length $1\text{m}$ is placed on a horizontal floor. A small block of mass $1\text{kg}$ is placed on top of the plank, at its right extreme end. The coefficient of friction between the plank and the floor is $0.5$ and that between the plank and block is $0.2$. If a horizontal force of $30\text{N}$ starts acting on the plank to the right, the time after which the block will fall off the plank is:- $(g=10{\text{m/s}}^2)$

• A. $\frac{2}{3}\text{s}$
• B. $1.5\text{s}$
• C. Block will never fall off the plank
• D. $\frac{4}{3}\text{s}$

Answer 1

The correct option is A $\frac{2}{3}\text{s}$

FBD of plank-block system is shown below: Block is slipping to the left w.r.t plank and plank is moving right w.r.t ground. Hence, kinetic friction acts in the direction opposite to relative motion.


Friction b/w plank & floor $f_1={\mu }_1\times 3g=15\text{N}$
Friction b/w plank & block $f_2={\mu }_2\times 1g=2\text{N}$

For block:
$1a_1=f_2$
For plank:
$2a_2=30-2-15=13$

Solving,
$a_1=2{\text{m/s}}^2$ & $a_2=6.5{\text{m/s}}^2$
$\Rightarrow a_{12}=a_1-a_2=2-6.5=-4.5{\text{m/s}}^2)$

When block falls off, $s_{12}=\frac{1}{2}{a}_{12}{t}^2$
$\Rightarrow -1=\frac{-9}{4}{t}^2$
or $t=\frac{2}{3}\text{s}$.