A body of mass $3\, kg$ is under a force, which causes a displacement in it is given by $S = \frac{{{t^3}}}{3}$ (in $m$). Find the work done by the force in first $2 seconds$ ............ $\mathrm{J}$
A body of mass $3\, kg$ is under a force, which causes a displacement in it is given by $S = \frac{{{t^3}}}{3}$ (in $m$). Find the work done by the force in first $2 seconds$ ............ $\mathrm{J}$
$2$
$3.8$
$5.2$
$24$
A body of mass $3\, kg$ is under a force, which causes a displacement in it is given by $S = \frac{{{t^3}}}{3}$ (in $m$). Find the work done by the force in first $2 seconds$ ............ $\mathrm{J}$
$S = \frac{{{t^3}}}{3}$
$dS = {t^2}dt$
$a = \frac{{{d^2}S}}{{d{t^2}}} = \frac{{{d^2}}}{{d{t^2}}}\left[ {\frac{{{t^3}}}{3}} \right] = 2t\;m/{s^2}$
Now work done by the force $W = \int\limits_0^2 {F.dS} = \int\limits_0^2 {ma.dS} $
$\int\limits_0^2 {3 \times 2t \times {t^2}dt} $$ = \int\limits_0^2 {6{t^3}dt} $= $\frac{3}{2}\left[ {{t^4}} \right]_0^2$
$= 24 \,J$
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