A sphere of density ρand radius a has a concentric cavity of...

A sphere of density $\rho$ and radius $a$ has a concentric cavity of radius $b$ as shown in the Fig. $7.13$. Gravitational potential energy as a function of $r$, where $r$ is the distance from the centre of the sphere. When $0<r<b$ (A) $U(r)=-2 \pi G \rho m\left(a^{2}-b^{2}\right)$. (B) $U(r)=\frac{-4 \pi G \rho m}{3 r^{2}}\left(a^{3}-b^{3}\right)$. (C) $U(r)=-\frac{2 \pi G \rho m}{3 r}\left(3 r a^{2}-2 b^{2}-r^{3}\right)$. (D) None of these.

A sphere of density $\rho$ and radius $a$ has a concentric cavity of radius $b$ as shown in the Fig. $7.13$.
Gravitational potential energy as a function of $r$, where $r$ is the distance from the centre of the sphere. When $0<r<b$
(A) $U(r)=-2 \pi G \rho m\left(a^{2}-b^{2}\right)$.
(B) $U(r)=\frac{-4 \pi G \rho m}{3 r^{2}}\left(a^{3}-b^{3}\right)$.
(C) $U(r)=-\frac{2 \pi G \rho m}{3 r}\left(3 r a^{2}-2 b^{2}-r^{3}\right)$.
(D) None of these.

Key Concepts

Gravitational Potential Energy

This concept involves the work done in assembling a mass configuration in a gravitational field or, equivalently, the energy possessed by a mass due to its position within a gravitational potential. In problems like the one given, it describes how potential energy depends on the spatial distribution of mass and is integral to calculating the binding energy of a system or the energy experienced by a test mass at a particular location.

Uniform Spherical Mass Distribution

A uniform spherical mass distribution is one in which mass is distributed evenly throughout the volume of a sphere. This concept is fundamental when dealing with gravitational fields because the symmetry of the sphere greatly simplifies calculations. The gravitational field within and outside such spheres can often be determined using simplified formulas derived from integration over the volume, and these principles underpin evaluations of gravitational potentials.

Newton's Shell Theorem

Newton's Shell Theorem states that a spherically symmetric shell of matter exerts no net gravitational force on a particle located anywhere inside the shell, and that for external points the gravitational effect is as if all the mass were concentrated at the center of the sphere. This theorem is critical in analyzing the gravitational potential in regions within and outside the mass distribution, and it plays a key role when a cavity is present in a spherical mass distribution, effectively allowing a subtraction of contributions.

Superposition Principle in Gravitational Fields

The superposition principle implies that the net gravitational potential due to a set of mass distributions is the algebraic sum of the potentials produced by each individual distribution considered separately. In the context of a sphere with a cavity, this principle enables us to compute the overall potential by adding the contributions from the complete sphere and then subtracting the gravitational potential effect of the missing (cavity) portion.

Effect of Cavities in Gravitational Potential

Introducing a cavity within a uniform mass distribution alters the gravitational potential because a portion of the mass, which would have contributed to the overall field, is now absent. The impact of the cavity must be accounted for by subtracting the gravitational contribution corresponding to the cavity’s mass. This approach, often leveraging the superposition principle and known results for uniform spheres, is essential for accurately determining the potential energy in modified mass configurations.

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