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Standard XII
Physics
Question

Two bodies of masses m and M are placed at a distance d apart. The gravitational potential at the position, where the gravitational field due to them is zero, is V. Then V is :
  1. V=−Gd(√m+√M)2
  2. V=Gd(m+M)
  3. V=−Gmd
  4. V=−GMd

A
V=Gd(m+M)
B
V=−GMd
C
V=−Gd(√m+√M)2
D
V=−Gmd
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Solution
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Let m and M be the masses.
d is the distance between body
Let x be the distance from the mass 'm'
at gravitational field is zero.
Net Force at x,F=Gmx2−GM(d−x)2
at x, F=0
Gmx2=GM(d−x)2

(d−x)2x2=Mm

(d−x)x=√M√m

x×(√M+√m)=√m×d

x=√md√M+√m

Potential at point x, V=−Gmx−GMd−x

Put x in V, we get
V=−Gm√md√M+√m−GM(d−√md√M+√m)

V=−G√md√M+√m−GM(√Md√M+√m)

V=−G√m(√M+√m)d−G√M(√M+√m)d

V=−Gd(√m+√M)2

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