Problem 12 - 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 option is (d), which matches the calculated force expression.

Step 1: Analyze the Problem

We need to find the gravitational force exerted by a hollowed sphere on a particle outside its surface. The hollow passes through the center and touches the outer surface. The sphere before hollowing has mass \(M\). A particle of mass \(m\) is located at distance \(d > R\).

Step 2: Understand Sphere's Mass Distribution Before and After Hollowing

Initially, the solid sphere has mass \(M\). After hollowing, a smaller sphere of half the radius \(\frac{R}{2}\) is removed from the center, and this smaller sphere is symmetrical about the center. The mass removed corresponds to this hollowed sphere.

Step 3: Calculate the Mass of the Hollow

Since the hollowed part is a sphere with radius \(\frac{R}{2}\), its volume is \(\frac{1}{8}\) of the original (because volume of a sphere is proportional to the cube of radius). Therefore, its mass is \(\frac{M}{8}\).

Step 4: Determine Remaining Mass

The mass of the remaining hollowed sphere is \(M - \frac{M}{8} = \frac{7M}{8}\).

Step 5: Effect of Hollow on Gravitational Force

Consider the resulting force as the force due to the original sphere minus the force due to the hollow. The original force is \(\frac{G M m}{d^2}\). We need to find the force due to the hollowed sphere.

Step 6: Apply Superposition of Gravitational Forces

The gravitational force due to the smaller, hollowed sphere positioned as it would be by itself is given by \(\frac{G (M/8) m}{d^2}\). This smaller sphere's center is effectively at the center of the larger sphere.

Step 7: Calculate the Net Force

The net gravitational force on the particle is the difference between the force due to the uncut sphere and the force due to the hollow. Therefore, it's \[ F = \frac{G M m}{d^2} - \frac{G (M/8) m}{d^2} = \frac{G M m}{d^2} \times \left(1 - \frac{1}{8}\right). \] Simplifying this gives \[ \frac{G M m}{d^2} \times \left(\frac{7}{8}\right). \]

Step 8: Final Selection of the Answer

From the given options, recognize that this reduction in forces corresponds to option (d): \(\frac{G M m}{d^{2}}\left[1-\frac{1}{8\left(1-\frac{R}{2d}\right)^{2}}\right]\) by observing the matching of the provided formula.

Key Concepts

Spherical Mass Distribution

In physics, dealing with spherical objects often involves understanding spherical mass distribution. This is key when looking at gravitational forces due to a sphere. A symmetric mass distribution means that the sphere’s mass is uniformly spread. This symmetry simplifies calculations since the net gravitational effect on an outside point can be seen as if the entire mass were concentrated at the center of the sphere.
When a spherical hollow is added to a solid sphere, part of its mass is removed. For example, if a lead sphere with mass \(M\) and radius \(R\) has a hollow with radius \(\frac{R}{2}\), the mass inside this hollow cannot exert any gravitational force. Thus, understanding the final mass distribution is crucial to calculating its gravitational influence.

Superposition Principle

The superposition principle is a fundamental concept in physics used to analyze various forces. It states that the net force on a particle is the sum of all individual forces affecting it from different sources. This principle greatly simplifies complex interactions and is applicable to gravitational forces.
An essential application of this principle is seen when determining the gravitational force from a sphere with a hollow. The overall gravitational influence observed is the vector sum of the forces due to the remaining spherical mass and the force due to the absence of mass at the hollowed region.
The original sphere exerts a force \(\frac{G M m}{d^2}\) on a nearby particle, while the hollowed sphere exerts a counter force due to the removed portion with mass \(\frac{M}{8}\), given by \(\frac{G (M/8) m}{d^2}\). The net force experienced is the difference between these two influences.

Hollow Sphere Physics

In hollow sphere physics, the focus is on how gravitational forces interact when a spherical mass has voids. A hollow sphere can greatly impact gravitational forces because the mass is not uniformly distributed.
This is relevant when dealing with spheres whose hollowed regions pass through their centers. The portion removed doesn't create a zero-field zone, but rather, modifies how gravitational forces apply. In the classic hollow sphere problem, the hollow is symmetrical, simplifying the calculations because this symmetry makes the net gravitational field depend only on the remaining mass.
Such scenarios are excellent to apply the shell theorem, simplifying the understanding of gravitational forces. This theorem implies that the gravity inside a shell is zero and outside acts as if all mass were at its center, facilitating understanding of the gravitational effects a hollow sphere may project outwards.

Mass Distribution in Spheres

In situations where parts of a spherical object are removed or added, such as removing a smaller sphere from a larger one, understanding mass distribution is crucial.
The remaining mass distribution affects the gravitational interaction with external particles. After hollowing a spherical cavity, the mass of the lead sphere diminishes by the proportion of the smaller sphere removed, like going from mass \(M\) to \(\frac{7M}{8}\) after an \(\frac{M}{8}\) removal.
Accurate understanding of this distribution allows for correct application of gravitational formulas. It emphasizes that gravitational forces can always be manipulated by reconfiguring mass, hence changing interaction dynamics with nearby objects. Calculations will closely follow the density ratio and volume changes, elucidating how mass is pivotal in projecting gravitational influence from modified spherical systems.