A body of mass 3 kg is under a constant force which causes a displacement $s$ in metres in it, given

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A body of mass 3 kg is under a constant force which causes a displacement $s$ in metres in it, given by the relation $s=\frac{1}{3} t^{2}$, where $t$ is in seconds. Work done by the force in 2 seconds is\

Professor PC Thomas Classes · Answer

(d) : $s=\frac{t^{2}}{3} ; \frac{d s}{d t}=\frac{2 t}{3} ; \quad \frac{d^{2} s}{d t^{2}}=\frac{2}{3} \mathrm{~m} / \mathrm{s}^{2}$ Work done, $W=\int F d s=\int m \frac{d^{2} s}{d t^{2}} d s$ $$ \begin{aligned} & =\int m \frac{d^{2} s}{d t^{2}} \frac{d s}{d t} d t=\int_{0}^{2} 3 	imes \frac{2}{3} 	imes \frac{2 t}{3} d t=\frac{4}{3} \int_{0}^{2} t d t \ & =\frac{4}{3}\left|\frac{t^{2}}{2}ight|_{0}^{2}=\frac{4}{3} 	imes 2=\frac{8}{3} \mathrm{~J} \end{aligned} $$