Question

A mass of 50 kg is placed at the centre of a uniform spherical shell of mass 100 kg and radius 50 m. If the gravitational potential at a point, 25 m from the centre is V. The value of V is:

• A. -60 G
• B. -20 G
• C. -4 G
• D. +2 G

Hint:

A key concept to remember is that the gravitational field inside a uniform spherical shell is zero, but the gravitational potential is not. It is constant and equal to the value at the surface. This is a common point of confusion.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We need to find the total gravitational potential at a point inside a spherical shell. The total potential is the sum of the potential due to the shell itself and the potential due to a point mass placed at its center.
Step 2: Key Formula or Approach:
The gravitational potential \(V\) at a point is calculated using the principle of superposition.
1. Potential due to a point mass M at distance r: \( V_p = -\frac{GM}{r} \)
2. Potential due to a uniform spherical shell of mass M\(_s\) and radius R:
- For a point inside the shell (\( r<R \)), the potential is constant and equal to the potential at the surface: \( V_{\text{shell, in}} = -\frac{GM_s}{R} \).
- For a point outside the shell (\( r>R \)), \( V_{\text{shell, out}} = -\frac{GM_s}{r} \).
Step 3: Detailed Explanation:
We are given:
Mass at the center, \( M_p = 50 \, \text{kg} \).
Mass of the spherical shell, \( M_s = 100 \, \text{kg} \).
Radius of the shell, \( R = 50 \, \text{m} \).
We need to find the potential at a distance \( r = 25 \, \text{m} \) from the center.
The total potential \( V_{\text{total}} \) at this point is the sum of the potential due to the point mass (\(V_p\)) and the potential due to the spherical shell (\(V_s\)).
\[ V_{\text{total}} = V_p + V_s \] Calculating Potential due to the Point Mass (\(V_p\)):
The point is at \( r = 25 \, \text{m} \) from the mass \( M_p = 50 \, \text{kg} \).
\[ V_p = -\frac{G M_p}{r} = -\frac{G \times 50}{25} = -2G \] Calculating Potential due to the Spherical Shell (\(V_s\)):
The point \( r = 25 \, \text{m} \) is inside the shell, since \( r<R \) (\( 25 \, \text{m}<50 \, \text{m} \)).
For any point inside a uniform spherical shell, the potential is constant and equal to the potential on its surface.
\[ V_s = -\frac{G M_s}{R} = -\frac{G \times 100}{50} = -2G \] Calculating Total Potential (\(V_{\text{total}}\)):
\[ V_{\text{total}} = V_p + V_s = (-2G) + (-2G) = -4G \] Step 4: Final Answer:
The value of the gravitational potential V at the given point is -4G.