Question

You have three balls, each of which is placed at a particular
location on the *x*-axis. Ball 1 has a mass of *m*,
and it is located at *x* = 0.

Ball 2 also has a mass of *m*, and it is located at
*x* = −*a*.

Ball 3 has a mass of 4*m*, and our goal is to determine
where it is located. We do know that that magnitude of the net
gravitational force experienced by ball 1, because of the other two
balls, is 2 Gm^{2}/a^{2}. Where is ball 3
located?

It turns out that there are two possible locations for ball 3.
Find the one closer to *x* = 0. Find both.

Answer #1

mass of ist ball = m

mass of 2nd ball = m

mass of 3rd ball =4m

net force on 1st ball,F1 = 2G*m^{2} /a^{2}

case-1 when ball 3rd is placed to the right of ist ball at a distance r.

F12= force on 1 due to 2 = Gm^{2} /
a^{2} ---------------------towards left

F13 = force on 1 due to 3 = G*4m^{2} /
r^{2} ----------------------------towards
left

net force on 1st ball F1= [G*4m^{2} / r^{2} ] -
[Gm^{2} / a^{2} ]

2G*m*a^{2} = [G*4m^{2} /
r^{2} ] - [Gm^{2} / a^{2} ]

on solving , r = 1.15 a

----------------------------------------------

case -II when ball placed towards left to ist ball.

net force F1= [G*4m^{2} / r^{2} ] +
[Gm^{2} / a^{2} ] ( both in same direction)

2Gm^{2} / a^{2} = [G*4m^{2} /
r^{2} ] + [Gm^{2} / a^{2} ]

on solving,r = 2a

distance closer to x=0 is ,r= 1.1 a

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