Question

A block of mass $4kg$ is kept on a rough horizontal surface with coefficient of friction ${\mu }_s=0.4$ and ${\mu }_k=0.3$ and a time varying horizontal force $F=4t$ applied on it, then the acceleration time graph of the particle is $(g=10{\text{m/s}}^2)$

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

As the force increases, the static friction self adjusts its value with $F$. But once $F$ goes beyond its limits, the block will start to accelerate.
Limiting value of frictional force is given as,
$(f_s{)}_{max}={\mu }_{s}mg(f_s{)}_{max}=0.4\times 4\times 10=16N$
Now, for $F=(f_s{)}_{max}4t=16\Rightarrow t=4s$
So the block starts moving after $4s$.

But as soon as the block starts moving, kinetic friction starts to act on block. Its value is given as:
$f_k={\mu }_{k}mg=0.3\times 4\times 10=12N$
So,
$t\le 4\Rightarrow F\le (f_x{)}_{max}\Rightarrow a=0$
$t>4\Rightarrow F>f_k,\Rightarrow a=\frac{F-f_k}{m}=\frac{4t-12}{4}=t-3\Rightarrow a>4-3\to a>1m/s^2$
$a=t-3$ suggests that graph is linear.

$\therefore$Graph in option $b$ represents correct information about the motion of the block.