Question

A particle of mass m is subjected to a force $\overrightarrow{F}=F_0\left[cos\right(t)\hat{i}+sin(t\left)\hat{j}\right]$. If initially (t = 0) the particle was at rest, the kinetic energy of the particle at time t is given by

• A. $\frac{F_0^2}{m}(1-cos2t)$
• B. $\frac{F_0^2}{m}(1-cost)$
• C. $\frac{F_0^2}{m}(1-sint)$
• D. $F_0^2\left(m\right)t$

Answer 1

The correct option is B $\frac{F_0^2}{m}(1-cost)$

$As,\overrightarrow{a}=\frac{\overrightarrow{F}}{m}=\frac{F_0}{m}\left[cos\right(t)\hat{i}+sin(t\left)\hat{j}\right]$
$\overrightarrow{v}={\int }_0^t\overrightarrow{a}dt=\frac{F_0}{m}\left[sin\right(t)\hat{i}-cos(t\left)\hat{j}\right]$
$\therefore kineticenergyattimet,$
$=\frac{1}{2}m(\overrightarrow{v}.\overrightarrow{v})=\frac{1}{2}\left[\frac{F_0^2}{m}\right(2-2cost\left)\right]=\frac{F_0^2}{m}(1-cost)$