Question

A force F acting on a particle depends upon displacement x as F $\propto$ $x^3$. The power delivered by force depends upon x as

• A. $x^{3/2}$
• B. $x^{7/2}$
• C. $x^5$
• D. $x^8$

Answer 1

The correct option is C $x^5$

Given,

$F\propto x^3$

$F=k{x}^3$

where, $k=$ proportionality constant.

Power, $P=F.v$. . . . . .(1)

$\frac{F}{m}=v\frac{dv}{dx}$

$\frac{k{x}^3}{m}=v\frac{dv}{dx}$

$\frac{k}{m}{\int }_0^x{x}^{3}dx={\int }_0^{v}vdv=\frac{v^2}{2}$

$\frac{k}{m}\frac{x^4}{4}=\frac{v^2}{2}$

$v^2=\frac{k}{2m}{x}^4$

$v=\sqrt{\frac{k}{2m}{x}^4}=\sqrt{\frac{k}{2m}}{x}^2$

From equation (1), power delivered is

$P=k{x}^3.\sqrt{\frac{k}{2m}}{x}^2=\sqrt{\frac{k^3}{2m}}{x}^5$

$P\propto x^5$

The correct option is C.