Question

A force $F$ acting on a particle depends on displacement as $F\propto x^n$. Then, power delivered by the force depends on $x$ as:

• A. $x^n$
• B. $x^{(2n+1)}$
• C. $x^{\frac{3n+1}{2}}$
• D. $x^{\frac{5n+2}{2}}$

Answer 1

The correct option is C $x^{\frac{3n+1}{2}}$

Given that,
$F\propto x^n\Rightarrow F=k{x}^n.....\left(1\right)$

Here, the force is a variable force as it depends on the position of the particle.

Hence, the acceleration can be written as $a=v\frac{dv}{dx}$
we can rewrite $\left(1\right)$ as,

$mv\frac{dv}{dx}=k{x}^n$

Seperating the variables and integrating on both sides, we get

$m\int vdv=k\int x^{n}dx$

$\Rightarrow \frac{v^2}{2}=\frac{k}{m}\frac{x^{n+1}}{(n+1)}+C$

$\Rightarrow v\propto x^{\frac{n+1}{2}}$

We know from the definition of power, $P=Fv$

i.e $P\propto [x^n\times x^{\frac{n+1}{2}}]\Rightarrow P\propto x^{\frac{3n+1}{2}}$

Thus, option (c) is the correct answer.