Question

The velocity-time graph in the figure given below shows the motion of a wooden block of mass $1\text{kg}$ which is given an initial push at $t=0$ along a horizontal table. Which of the following statements is/are correct ?

• A. The coefficient of friction between the block and the table is $0.1$
• B. The coefficient of friction between the block and the table is $0.2$
• C. If the table has half of its present roughness, the time taken by the block to complete the journey is $4\text{s}$
• D. If the table has half of its present roughness, the time taken by the block to complete the journey is $8\text{s}$

Answer 1

Answer: (A), (D)

If the table has half of its present roughness, the time taken by the block to complete the journey is $8\text{s}$

From the $v-t$ graph:
$\frac{dv}{dt}=\text{Slope}=\frac{(0-4)}{(4-0)}=-1$
$\Rightarrow a=\frac{dv}{dt}=-1{\text{m/s}}^2$

Negative acceleration is a result of friction acting on the block in opposite direction.
$\therefore f=m|a|=1\times 1=1\text{N}$ in opposite direction to the motion

$\because f=\mu N=\mu mg$
$\Rightarrow \mu =\frac{1}{10}=0.1$

If surface roughness becomes half; $\mu =0.05$
$\therefore f=0.05\times 10=0.5\text{N}$
and $|a|=\frac{0.5}{1}=0.5{\text{m/s}}^2$

For block to complete its journey: (Final velcoity is zero)
$\Rightarrow u=4\text{m/s},v=0,a=-0.5{\text{m/s}}^2$
From $v=u+at$
$0=4-\left(0.5\right)t$
$\therefore t=8\text{s}$