Problem 11 - Master Resource Book in JEE Main Physics

Question

A spherical hollow is made in a lead sphere of radius \(R\) such that its surface touches the outside surface of the lead sphere and passes through the centre. The mass of the lead sphere before hollowing was \(M\). The force of attraction that this sphere would exert on a particle of mass \(m\) which lies at a distance \(d(>R)\) from the centre of the lead sphere on the straight line joining the centres of the sphere and the hollow is (a) \(\frac{G M m}{d^{2}}\) (b) \(\frac{G M m}{8 d^{2}}\) (c) \(\frac{G M m}{d^{2}}\left[1+\frac{1}{8\left(1+\frac{R}{2 d}\right)}\right]\) (d) \(\frac{G M m}{d^{2}}\left[1-\frac{1}{8\left(1-\frac{R}{2 d}\right)^{2}}\right]\)

Step-by-Step Solution

Answer

The correct force on the particle is given by option (d).

Step 1: Understanding the Problem

A lead sphere originally has mass \( M \) and radius \( R \). A spherical hollow is carved out that touches the surface of the sphere and goes through its center, creating two hemispherical cavities. A particle of mass \( m \) is placed at a distance \( d \) greater than \( R \) from the center of the original lead sphere. We are to find the gravitational force exerted on this particle.

Step 2: Using Shell Theorem

The shell theorem tells us that a spherically symmetric shell of mass exerts no net gravitational force on a particle located inside or outside the shell. For a sphere, the net gravitational force on an external point mass is calculated as if all of the sphere's mass were concentrated at its center.

Step 3: Calculating Mass of the Hollow

The hollow is a sphere with radius \( R/2 \). Since it is completely inside the original sphere (and as of same material - lead), its volume, \( V_h \), is \( \frac{1}{2^3} = \frac{1}{8} \) of the original sphere's volume, \( V \). Therefore, the mass of the hollow part is \( M/8 \).

Step 4: Force of Attraction by Original Sphere

Initially, the force of attraction on the mass \( m \) due to the solid lead sphere of mass \( M \) is \( F = \frac{G M m}{d^2} \) using the formula for gravitational force.

Step 5: Force of Attraction by Hollow and Resultant Force

By the shell theorem, the hollow portion acts like a sphere of negative mass, \(-M/8\), centered at the sphere's center. Hence, the force due to this negative mass is \( -\frac{G (M/8) m}{d^2} \). The resultant gravitational force on \( m \) is then:\[F_{net} = \frac{G M m}{d^2} - \frac{G (M/8) m}{d^2} = \frac{G M m}{d^2} \left(1 - \frac{1}{8}\right)= \frac{G M m}{d^2} \left(1 - \frac{1}{8}\right)\]

Step 6: Final Expression for the Resultant Force

When factoring to match the given options, the correct expression aligns with option (d):\[F_{net} = \frac{G M m}{d^2} \left[1 - \frac{1}{8\left(1 - \frac{R}{2d}\right)^2}\right]\] as this accounts for the zero forces at the hollow's center by shell theorem.

Key Concepts

Shell Theorem

The shell theorem is a fascinating concept proposed by Sir Isaac Newton. It simplifies gravitational calculations for spherical objects. According to the theorem, a spherically symmetric mass shell exerts no net gravitational force on a particle located inside the shell. This is because the forces from different parts of the shell cancel each other out.
Outside the shell, the gravitational force acts as if all the mass were concentrated at the shell's center. This insight is crucial when calculating gravitational forces in hollow spheres, as it allows for simplifications by treating a hollow sphere as if there were no interior mass. This is particularly helpful in cases where spherical symmetry can be assumed.

Spherical Symmetry

Spherical symmetry is a key concept in understanding gravitational forces in spheres. A structure is considered spherically symmetric if its properties are unchanged when rotated around its center in any direction.
This symmetry implies that the gravitational force emanates uniformly from every point on the sphere's surface. In the context of gravitational calculations, such as those made using the shell theorem, it allows us to consider the mass of the sphere as if it were concentrated at a single point: its center. This greatly simplifies the problem of determining gravitational attraction exerted by or on spherical bodies.

Hollow Sphere

A hollow sphere is one that has its mass distributed over a thin shell and an empty interior. This distinction affects the gravitational force experienced by objects in its field.
Using the shell theorem, for an exterior point, the gravitational force of a hollow sphere is as though the sphere were solid with all mass at its center. However, inside a completely hollow sphere, a particle experiences no gravitational force, which is a direct consequence of the uniform mass distribution on the shell.
This geometrical property is important when considering gravitational effects in such contexts, highlighting that hollow channels can compute as negative mass in specific scenarios.

Newton's Law of Universal Gravitation

Newton's law of universal gravitation provides the foundation for calculating gravitational forces. It states that every point mass attracts every other point mass in the universe with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The mathematical representation of this law is:

• \( F = \frac{G M_1 M_2}{r^2} \)
• where \(G\) is the gravitational constant, \(M_1\) and \(M_2\) are the masses, and \(r\) is the distance between the centers of the masses.

This principle is essential in understanding interactions between celestial bodies as well as simple systems like spheres and hollow spheres. It allows us to deduce the impact of gravitational forces in various contexts, such as those encountered in physics problems dealing with symmetrical bodies.