Question:

Four spheres each of mass m form a square of side d (as shown in figure). A fifth sphere of mass M is situated at the centre of square. The total gravitational potential energy of the system is:
Four spheres each of mass m form a square of side d

Updated On: Dec 9, 2024
  • \(−\frac{Gm}{d}\bigg[\bigg(4+\sqrt2\bigg)m+4\sqrt2M\bigg]\)
  • \(−\frac{Gm}{d}\bigg[\bigg(4+\sqrt2\bigg)M+4\sqrt2m\bigg]\)
  • \(−\frac{Gm}{d}\bigg[3m^2+4\sqrt2M\bigg]\)
  • \(−\frac{Gm}{d}\bigg[6m^2+4\sqrt2M\bigg]\)
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The Correct Option is A

Solution and Explanation

Total gravitational potential energy

=\(−{\frac{4GMm}{\frac{d}{\sqrt2}}+\frac{4Gm^2}{d}+\frac{2Gm^2}{\sqrt2d}\\}\)

=\(−\frac{Gm}{d}\{{M4\sqrt2+(4+\sqrt2)m}\}\)

=\(−\frac{Gm}{d}\{{4\sqrt2M+(4+\sqrt2)m}\}\)

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Concepts Used:

Gravitational Potential Energy

The work which a body needs to do, against the force of gravity, in order to bring that body into a particular space is called Gravitational potential energy. The stored is the result of the gravitational attraction of the Earth for the object. The GPE of the massive ball of a demolition machine depends on two variables - the mass of the ball and the height to which it is raised. There is a direct relation between GPE and the mass of an object. More massive objects have greater GPE. Also, there is a direct relation between GPE and the height of an object. The higher that an object is elevated, the greater the GPE. The relationship is expressed in the following manner:

PEgrav = mass x g x height

PEgrav = m x g x h

Where,

m is the mass of the object,

h is the height of the object

g is the gravitational field strength (9.8 N/kg on Earth) - sometimes referred to as the acceleration of gravity.