Question

8.
Two point objects of mass 2x and 3x are separated by a distance r. Keeping the distance fixed, how much
mass should be transferred from 3x to 2x, so that gravitational force between them becomes maximum?
a) x/4
b) x/3
c) x/2
d) 2x/3
pls answer logically otherwise I will reject ur answer. Thank you ​

Answer 1

Answer

c)x/2

Explanation:

Given

Mass of particle(1)=2x

Mass of particle (2)=3x

Fixed Distance between them=r

Let assume that  m unit of mass is transferred from 3x to 2x.

Then according to Newtons gravitational Law Law the force between them is given by

$F=\dfrac{G(3x-m)(2x+m)}{r^2}$

Since here m is variable So Differentiating the force with respect to m and putting it to zero we have

$\dfrac{dF}{dm}=0\\\dfrac{d}{dm}\left (\dfrac{G(3x-m)(2x+m)}{r^2} \right )=0\\-2x-m+3x-m=0\\m=x/2$

Now checking if force is maximum or not

$\dfrac{dF}{dm}=\dfrac{G(x-2m)}{r^2}\\\\\dfrac{d^2F}{dm^2}=\dfrac{-2G}{r^2}$

since G and r are positive constants

$\dfrac{d^2F}{dm^2}=\dfrac{-2G}{r^2}<0$

since the second derivative is negative at m=x/2 so the force is maximum at m=x/2