Question

A particle is moving along x-axis under the action of a force, F which varies with its position (x) as $F\propto \frac{1}{\sqrt[4]{4x}}$. Find the variation of power due to this force with respect to x.

Answer 1

$F\alpha \frac{1}{x^{1/4}}$

$F=k{x}^{1/4}$

$ma=k{x}^{-1/4}$

$m\left(v\frac{dv}{dx}\right)=k{x}^{1/4}$

$\int vdv=\frac{k}{m}\int x^{-1/4}dx$

$\frac{v^2}{2}=-\frac{k}{m}{x}^{3/4}$

$v=-\frac{2k}{m}{x}^{3/4}$

Power $=F.v$

Power $\alpha \frac{1}{x^{1/4}}\times x^{3/4}\alpha \sqrt{x}$