Question

1 . Is external force always necessary for work to be done ?
2. Why is work done by a varying force
during a round trip not necessarily zero?
3. The position (x) of a particle of mass 1kg moving along x-axis at time t is given by (x=1by 2 t²⁾ metre. Find the work done by force acting on it in time interval from t=0 to t=3s.

Answer 1

$$\begin{gathered} 1.\mathrm{Workdone}\mathrm{W}=\mathrm{F}\mathrm{S}\mathrm{cos\theta }. \\ \mathrm{If}\mathrm{External}\mathrm{force}\mathrm{F}=0\mathrm{then}\mathrm{W}=0\mathrm{because}\mathrm{internal}\mathrm{force}\mathrm{does}\mathrm{count}\mathrm{to}\mathrm{workdone}. \\ 2.\mathrm{Workdone}\mathrm{by}\mathrm{variable}\mathrm{force}\mathrm{for}\mathrm{the}\mathrm{round}\mathrm{may}\mathrm{or}\mathrm{may}\mathrm{not}\mathrm{be}\mathrm{zero}.\mathrm{It}\mathrm{depends}\mathrm{on} \\ \mathrm{the}\mathrm{resultant}\mathrm{force}.\mathrm{In}\mathrm{case}\mathrm{resultant}\mathrm{force}\mathrm{of}\mathrm{variable}\mathrm{forces}\mathrm{is}\mathrm{zero}\mathrm{then}\mathrm{workdone}\mathrm{by} \\ \mathrm{variable}\mathrm{force}\mathrm{is}\mathrm{zero}\mathrm{and}\mathrm{on}\mathrm{the}\mathrm{other}\mathrm{hand}\mathrm{if}\mathrm{resultant}\mathrm{force}\mathrm{is}\mathrm{non}\mathrm{zero}\mathrm{then}\mathrm{workdone} \\ \mathrm{by}\mathrm{variable}\mathrm{force}\mathrm{is}\mathrm{also}\mathrm{non}\mathrm{zero}. \\ 3.\mathrm{x}=\frac{1}{2{\mathrm{t}}^2}.....\left(\mathrm{i}\right) \\ \frac{\mathrm{dx}}{\mathrm{dt}}=-\frac{2}{2{\mathrm{t}}^3}\mathrm{and}\frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{dt}}^2}=\mathrm{a}=-\frac{-3}{{\mathrm{t}}^4}=\frac{3}{{\mathrm{t}}^4} \\ \mathrm{from}\left(\mathrm{i}\right), \\ {\mathrm{t}}^2=\frac{1}{2\mathrm{x}} \\ {\mathrm{t}}^4=\frac{1}{4{\mathrm{x}}^2} \\ \mathrm{a}=12{\mathrm{x}}^2 \\ \mathrm{Therefore}, \\ \mathrm{W}={\int }_0^3\mathrm{Fdx}={\int }_0^3\mathrm{ma}\mathrm{dx}={\int }_0^{3}1\times 12{\mathrm{x}}^2\mathrm{dx} \\ \mathrm{W}=\frac{12}{3}{\left[{\mathrm{x}}^3\right]}_0^3\mathrm{J} \\ \mathrm{W}=4\times 27=108\mathrm{J} \end{gathered}$$