Problem 64 - Master Resource Book in JEE Main Physics

Question

A particle of mass \(m\) is placed at the centre of a uniform spherical shell of mass \(3 \mathrm{~m}\) and radius \(R\). The gravitational potential on the surface of the shell is (a) \(-\frac{G m}{R}\) (b) \(-\frac{3 \mathrm{Gm}}{R}\) (c) \(-\frac{4 G m}{R}\) (d) \(-\frac{2 G m}{R}\)

Step-by-Step Solution

Answer

The answer is (c) \(-\frac{4Gm}{R}\).

Step 1: Understanding the setup

We have a particle of mass \( m \) placed at the center of a spherical shell. The shell itself has a mass of \( 3m \) and a radius \( R \). We need to calculate the gravitational potential on the surface of this spherical shell.

Step 2: Gravitational potential due to the shell

Since the particle is inside a uniform spherical shell, the gravitational potential due to the spherical shell at any point inside, including the center, is constant. The potential at the surface due to the spherical shell itself is given by:\[ V_{shell} = -\frac{G \times 3m}{R} \]

Step 3: Gravitational potential due to the particle

For a point mass, the gravitational potential at a distance \( R \) from the mass is given by:\[ V_{particle} = -\frac{G m}{R} \]

Step 4: Total gravitational potential on the shell's surface

To find the total gravitational potential on the surface of the shell, we sum the potentials due to the shell and the particle:\[ V_{total} = V_{shell} + V_{particle} = -\frac{3Gm}{R} - \frac{Gm}{R} = -\frac{4Gm}{R} \]

Step 5: Conclusion: Choosing the correct option

The total gravitational potential on the surface of the shell is \(-\frac{4Gm}{R}\), which corresponds to option (c).

Key Concepts

Uniform Spherical Shell

A uniform spherical shell is essentially a hollow ball, where the mass is evenly distributed across its surface. Such a configuration has some rather interesting properties regarding gravitational forces and potentials.

Here are a few key characteristics:

• The gravitational potential inside the shell is uniform, meaning it doesn't change at different points inside. This is due to the symmetry and mass distribution on the shell's surface.
• When a point mass is placed anywhere inside the shell, including at its center, the shell's gravitational influence will be the same throughout its interior.
• Outside the shell, it behaves like a point mass situated at its center, making calculations less complex when dealing with distances from the center.

Understanding these properties helps explain why adding a mass at the center does not alter the uniform potential inside, apart from the contribution from the mass itself.

Gravitational Field

The gravitational field is a way to express how massive bodies exert forces on one another. It can be thought of as the "reach" of a body's gravity, influencing other objects around it. Here's how it works in the case of a spherical shell:

• Inside a uniform spherical shell, the gravitational field is zero. This means that any object placed inside experiences no net gravitational force from the shell. The forces from the surrounding mass perfectly cancel out.
• Outside the shell, the gravitational field can be calculated just as if all the shell's mass were concentrated at its center. This simplifies computations, acting exactly like a point mass scenario.
• The strength of a gravitational field is measured in terms of force per mass, usually expressed in Newtons per kilogram (N/kg).

By understanding the gravitational field, we perceive how the shell’s mass affects other bodies within and without its bounds.

Point Mass

A point mass is a hypothetical object with mass but zero size, used to simplify the calculation of gravitational influence. In this problem, the particle placed at the center of the shell can be treated as a point mass. Here’s why a point mass is important:

• A point mass simplifies mathematical models by focusing solely on the mass’s location and ignoring volumes and shapes.
• Gravitational potential due to a point mass is calculated using the formula:\[ V = -\frac{Gm}{r} \]where \( V \) is the potential, \( G \) is the gravitational constant, \( m \) is the mass, and \( r \) is the distance from the mass.
• In scenarios involving multiple gravitational sources, like this one, the point mass concept helps in superimposing influences to find total potentials.

Considering the particle at the center as a point mass allows us to easily add its potential to that of the shell to achieve the total gravitational potential on the shell’s surface.