Question

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

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

Answer 1

Answer: (A)

$x^0$

Answer 2

As $F\propto x^{-1/3}$ $\therefore$ Acceleration, $a \propto x^{-1/3}$ $a = \frac{dv}{dt} = \frac{dv}{dt } \frac{dx}{dx} = \frac{dx}{dt} \frac{dv}{dx} = v \frac{dv}{dx}$ i.e., $v \frac{dv}{dx}\propto x^{-1/3}$ Integrating both sides, we get $v^{2}\propto x^{2/3}$ or$v\propto x^{1/3}$ As Power, $P = Fv$ $\therefore P\propto\left(x^{-1/3}\right)\left(x^{1/3}\right)$ or $P\propto x^{0}$ Power will be independent of $x$.