Question

Two metallic spheres of mass $M$ are suspended by two strings each of length $L$ . The distance between the upper ends of strings is $L$. The angle which the strings will make with the vertical due to mutual attraction of sphere is (if each mass horizontally moved by a distance of $\dfrac{L}{4}$ due to mutual attraction)
$(A){\tan ^{ - 1}}\left[ {\dfrac{{GM}}{{gL}}} \right]$
$(B){\tan ^{ - 1}}\left[ {\dfrac{{GM}}{{2gL}}} \right]$
$(C){\tan ^{ - 1}}\left[ {\dfrac{{GM}}{{g{L^2}}}} \right]$
$(D){\tan ^{ - 1}}\left[ {\dfrac{{4GM}}{{g{L^2}}}} \right]$

Answer 1

All physical things that come into contact with one another can exert forces on one another. Depending on the types of objects in touch, we call these contact forces by different names.
The force tension is what we call it when one of the things applying the force is a rope, string, chain, or cable.
When you tug on anything with a rope, it stretches slightly (often imperceptibly). This stretch causes the rope to be taut (i.e. under strain), allowing it to transfer force from one side of the rope to the other, like how a stretched spring pulls on objects attached to it.

Complete step-by-step solution:
Given,
$M$= mass of metallic spheres
$L$ = length of the strings
$L$ = distance between upper ends of the strings
The formula used for calculating the gravitational force between two objects :
$F = \dfrac{{GMm}}{{{R^2}}}$
Now, for the given question
$F = \dfrac{{GM.M}}{{{L^2}}}$
$T\sin \theta = F$
$T\cos \theta = Mg$
$\tan \theta = \dfrac{F}{{Mg}}$
$\tan \theta = \dfrac{{\dfrac{{G{M^2}}}{{{L^2}}}}}{{Mg}}$
So,
$\theta = {\tan ^{ - 1}}\left( {\dfrac{{GM}}{{{L^2}{g^{}}}}} \right)$

Note: The magnitude of the force is proportional to the masses of the two objects and inversely proportional to their distance.
A mass attracts another mass; the magnitude of the force is proportional to the masses of the two objects and inversely proportional to the square of the distance between them.
The force is inversely proportional to the square of their mass centers and proportionate to the product of their masses. An inverse-square law is what this is known as.
Coulomb's law of electrical forces, which is used to quantify the amount of the electrical force-generating between two charged things, is similar to Newton's law of gravitation.