Question

A particle is moving along the X-axis. The force acting on it is proportional to x−1/3,where x is the displacement of the particle from the origin. If the instantaneous power delivered by that force to the particle is proportional to xn, the value of n is

A) 2/3
B) 1/3
C) zero
D) -2/3

Answer 1

Force acting on the particle is directly proportional to x^(-⅓)

We can say that :

$F \: \propto \: {x}^{ - \frac{1}{3} }$

Introducing a constant :

$F \: = \: k{x}^{ - \frac{1}{3} }$

Acceleration of the object shall be force divided by the mass :

$a = \dfrac{F}{m} \: = \: \dfrac{k}{m}{x}^{ - \frac{1}{3} }$

$= > v \dfrac{dv}{dx} = \dfrac{k}{m} {x}^{ - \frac{1}{3} }$

Let k/m be another constant c :

$= > v \dfrac{dv}{dx} = c {x}^{ - \frac{1}{3} }$

$= > v \: dv = c \: {x}^{ - \frac{1}{3} } dx$

Integrating on both sides :

$= > \displaystyle \int \: v \: dv = c \: \int{x}^{ - \frac{1}{3} } dx$

$= > \dfrac{ {v}^{2} }{2} = \dfrac{3c}{2} \: {x}^{ \frac{2}{3} }$

$= > v = \sqrt{ 3c } \: {x}^{ \frac{1}{3} }$

$= > v \: \propto \: {x}^{ \frac{1}{3} }$

We know that Instantaneous Power is P :

$P = F \times v$

$= > P \: \propto \: {x}^{ - \frac{1}{3} } \times {x}^{ \frac{1}{3} }$

$= > P \: \propto \: {x}^{ 0}$

So final answer is :

Value of n is zero .