Gravitation Part 1

Given, total mass of body is 2M.
Potential energy is maximum at

M

m

=

x

1

The arrangement of masses to form a square is shown in diagram. The gravitational potential energy of a body is given
U=−

GMm

r

For the given system, the potential energy will be UT=−

4‌Gm(M−m)

d

−

Gm2

√2d

−

G(M−m)2

√2d

For maximum potential energy,

dUT

dm

=0
−

4G

d

[M−2m]−

G

√2d

[2m]
−

G

√2d

[2(M−m)×−1]=0
⇒4M−8m+√2m=√2(M−m)
(4−√2)M=(8−2√2)m

M

m

=

2(4−√2)

4−√2

=2
Comparing it with the given value, we get x = 2
Thus, potential energy will be maximum when x = 2.