Question

A force $F$ acting on a body depends on its displacement $S$ as $F\propto S^{-1/3}$. The power delivered by $F$ will depend on displacement as

• A. $S^{2/3}$
• B. $S^{1/6}$
• C. $S^0$
• D. $S^{-5/3}$

Answer 1

Answer: (C)

$S^0$

Given,

$F\propto S^{-1/3}$

$F=k{S}^{-1/3}$. . . . . . . .(1)

where, $k=constant$

Force,

$F=ma=mv\frac{dv}{dS}$. . . . . . .(2)

from equation (1) and (2), we get

$mv\frac{dv}{dS}=k{S}^{-1/3}$

$\int vdv=\frac{k}{m}\int S^{-1/3}dS$

$\frac{v^2}{2}=\frac{k}{m}\times \frac{3}{2}{S}^{2/3}$

$v=\sqrt{\frac{k}{m}}{S}^{1/3}$

The power delivered by the force would be

$P=Fv$

$P=k{S}^{-1/3}.\sqrt{\frac{k}{m}}{S}^{1/3}=k\sqrt{\frac{k}{m}}{S}^0$

$P\propto S^0$

The correct option is C.