Question

A block of mass $1.5\text{kg}$ rests on a rough horizontal surface. A horizontal force applied to the block increases uniformly from $0$ to $15N$ in $5$ seconds. The velocity (in $m/s$) of the block after $5$ seconds is (given ${\mu }_s=0.6,{\mu }_k=0.5andg=10{m/s}^2$

Answer 1

As force is increasing linearly from $0$ to $15N$ in $5$ seconds.
Applied force $F=mt$ where $m=\frac{15}{5}=3$
$\Rightarrow F=3t$
Limiting friction $f_{max}={\mu }_{s}mg=0.6\times 1.5\times g$
For the block to move, $F\ge f_{max}$
Hence $3t\ge 0.6\times \frac{3}{2}\times 10\Rightarrow t\ge 3\text{sec}$
i.e., the blocks starts to move at $t=3\text{sec}$ and $u=0$ (at $3$ sec)
Net force for $3\le t\le 5$ sec is
$3t-{\mu }_{k}mg=ma\Rightarrow 3t-0.5\times 1.5\times 10=1.5\frac{dv}{dt}$
$\Rightarrow \underset{0}{\overset{v}{\int }}dv=\underset{3}{\overset{5}{\int }}(2t-5)dt$
$\Rightarrow v={[2\frac{t^2}{2}-5t]}_3^5=6m/s$