Question

A force $F$ acting on a body depends on its displacement $x$ as $F\propto x^n$. The power delivered by $F$ will be independent of $x$ if $n$ is:

• A. $\frac{1}{3}$
• B. $-\frac{1}{3}$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$

Answer 1

Answer: (B)

$-\frac{1}{3}$

Here, acceleration $=v\frac{dv}{dx}=k{x}^n,$ where $k$ is a constant.

$\therefore \frac{v^2}{2}=k\left(\frac{x^{n+1}}{n+1}\right)$

$\therefore v\propto x^{\left(\frac{n+1}{2}\right)}$

Power $P=Fv\propto \left(x^n\right)\left(x^{\frac{n+1}{2}}\right)=x^{\left(\frac{3n+1}{2}\right)}$

For $P$ to be independent of $x$, $\frac{3n+1}{2}=0$

or $n=-\frac{1}{3}$