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:

• A. \(−\frac{Gm}{d}\bigg[\bigg(4+\sqrt2\bigg)m+4\sqrt2M\bigg]\)
• B. \(−\frac{Gm}{d}\bigg[\bigg(4+\sqrt2\bigg)M+4\sqrt2m\bigg]\)
• C. \(−\frac{Gm}{d}\bigg[3m^2+4\sqrt2M\bigg]\)
• D. \(−\frac{Gm}{d}\bigg[6m^2+4\sqrt2M\bigg]\)

Answer 1

The Correct Option is A

To find the total gravitational potential energy of the system, we need to consider the potential energy between each pair of masses. The system consists of four spheres of mass \( m \) at the corners of a square of side \( d \), and a fifth sphere of mass \( M \) at the center.

1. Calculate the potential energy between the spheres at the corners:
   • Each pair of adjacent spheres at the corners contributes potential energy: \(-\frac{Gm^2}{d}\).
   • There are 4 pairs of adjacent corners: 4 \times -\frac{Gm^2}{d} = -\frac{4Gm^2}{d}.
   • Each diagonal pair (pairs across the diagonal of the square) has a separation of \(\sqrt{2}d\) and contributes potential energy: \(-\frac{Gm^2}{\sqrt{2}d}\).
   • There are 2 diagonal pairs: 2 \times -\frac{Gm^2}{\sqrt{2}d} = -\frac{2\sqrt{2}Gm^2}{2d}\).
2. Calculate the potential energy between the central sphere and the corner spheres:
   • The distance from the center to any corner is \(\frac{d}{\sqrt{2}}\).
   • Potential energy for each central-corner sphere pair: \(-\frac{GmM}{d/\sqrt{2}} = -\frac{\sqrt{2}GmM}{d}\).
   • With 4 corner spheres, the total potential energy: 4 \times -\frac{\sqrt{2}GmM}{d} = -\frac{4\sqrt{2}GmM}{d}.
3. Add all the potential energies:
   • Total potential energy: \[ U = -\frac{4Gm^2}{d} - \frac{2\sqrt{2}Gm^2}{d} - \frac{4\sqrt{2}GmM}{d} \] \[ = -\frac{G}{d} \left[(4 + \sqrt{2})m^2 + 4\sqrt{2}mM\right] \]

The correct option corresponds to: \(-\frac{Gm}{d}\bigg[\bigg(4+\sqrt2\bigg)m+4\sqrt2M\bigg]\).