Question

Two point masses m and 4m are separated by a distance d on a line at third point mass m not used to be placed at a point on the line such that the net gravitational force on its is zero the distance of the point from the mass m each

Answer 1

Hence d/3 will be the distance from mass m. Answer is as follows....

Answer 2

Given, two point masses of m and 4m are separated by distance'd' on line.

At third point mass 'm' is not placed at a point on the line. Therefore the gravitational force is considered to be zero at that point.

Therefore,

                  $\quad \cfrac { Gm }{ { x }^{ 2 } } \quad =\quad \frac { G\left( 4m \right) }{ { \left( r-x \right)}^{ 2 } }$

                 $\frac { 1 }{ x } \quad =\quad \frac { 2 }{ r-x }$

                 $r-x\quad =\quad 2x\\x\quad =\quad \frac { r }{ 3 }$

 

Consider the point P is at the distance $\quad \frac { r }{ 3 }$ from mass 'm' and at the distance $\frac { 2r }{ 3 }$ from mass '4m'

                 $\quad V\quad =\quad \frac { -GM }{ \frac { r }{ 3 } } -\frac { g\left( 4m \right) }{ \frac { 2r }{ 3 } }$

                 $=\quad \frac { -9GM }{ r } \quad$

   

Therefore, at point 'P' on the line where the net gravitational force is zero; is at the distance of the point from the mass m each is $\frac { -9GM }{ r } \quad$