A solid sphere of mass M and radius R is surrounding by a spherical shell of same mass M and rad

step 1 In this question we will learn about some basic concepts of the gravitation. The question is the

A solid sphere of mass M and radius R is surrounding by a...

A solid sphere of mass $\mathrm{M}$ and radius $\mathrm{R}$ is surrounding by a spherical shell of same mass $\mathrm{M}$ and radius $2 \mathrm{R}$ as shown. A small particle of mass $m$ is released from rest from a height $(\mathrm{h}<<\mathrm{R})$ above the shell there is a hole in the shell. (i) in what time will it enter the hole at $\mathrm{A}$ (A) $\left[\left(2 \sqrt{\mathrm{hR}}^{2}\right) /(\mathrm{GM})\right]$ (B) $\left.\sqrt{[}\left(2 \mathrm{hR}^{2}\right) /(\mathrm{GM})\right]$ (D) None of these(ii) What time will it take to move from $\mathrm{A}$ to $\mathrm{B}$ ? $(\mathrm{A})=\left[\mathrm{R}^{2} /(\sqrt{\mathrm{GMh}})\right]$ (B) $>\left[\mathrm{R}^{2} /(\sqrt{\mathrm{G} M h})\right]$ $(\mathrm{C})<\left[\mathrm{R}^{2} /(\sqrt{\mathrm{G} M h})\right]$ (D) None of these (iii) with what approximate speed will it collide at $\mathrm{B}$ ? (A) $\sqrt{[}(2 \mathrm{GM}) / \mathrm{R}]$ (B) $\sqrt{[}(3 \mathrm{GM}) /(2 \mathrm{R})]$ (C) $\sqrt{[}(\mathrm{GM}) /(2 \mathrm{R})]$ (D) $\sqrt{(\mathrm{GM} / \mathrm{R})}$

A solid sphere of mass $\mathrm{M}$ and radius $\mathrm{R}$ is surrounding by a spherical shell of same mass $\mathrm{M}$ and radius $2 \mathrm{R}$ as shown. A small particle of mass $m$ is released from rest from a height $(\mathrm{h}<<\mathrm{R})$ above the shell there is a hole in the shell.
(i) in what time will it enter the hole at $\mathrm{A}$
(A) $\left[\left(2 \sqrt{\mathrm{hR}}^{2}\right) /(\mathrm{GM})\right]$
(B) $\left.\sqrt{[}\left(2 \mathrm{hR}^{2}\right) /(\mathrm{GM})\right]$
(D) None of these(ii) What time will it take to move from $\mathrm{A}$ to $\mathrm{B}$ ?
$(\mathrm{A})=\left[\mathrm{R}^{2} /(\sqrt{\mathrm{GMh}})\right]$
(B) $>\left[\mathrm{R}^{2} /(\sqrt{\mathrm{G} M h})\right]$
$(\mathrm{C})<\left[\mathrm{R}^{2} /(\sqrt{\mathrm{G} M h})\right]$
(D) None of these
(iii) with what approximate speed will it collide at $\mathrm{B}$ ?
(A) $\sqrt{[}(2 \mathrm{GM}) / \mathrm{R}]$
(B) $\sqrt{[}(3 \mathrm{GM}) /(2 \mathrm{R})]$
(C) $\sqrt{[}(\mathrm{GM}) /(2 \mathrm{R})]$
(D) $\sqrt{(\mathrm{GM} / \mathrm{R})}$

Key Concepts

Free Fall Motion Under Gravity

Free fall motion describes how objects accelerate solely due to gravity when other forces (like air resistance) are negligible. In problems like this, analyzing the time of descent and displacement involves applying the equations of motion under a gravitational field, particularly when the acceleration can sometimes be approximated as constant or varies predictably.

Energy Conservation in Gravitational Fields

Energy conservation is a fundamental principle used to relate the gravitational potential energy lost by the particle to its kinetic energy gained during its fall. This approach is critical for determining quantities like speed upon collision and helps simplify the analysis of motion by avoiding complex force integrations.

Composite Gravitational Systems

In a system comprising multiple symmetric mass distributions, such as a solid sphere surrounded by a spherical shell, the overall gravitational field at any point is determined by vectorially adding the contributions from each component. Understanding this superposition is key to correctly setting up the forces and energy equations that govern the particle’s motion.

Gravitational Field of a Uniform Sphere

For a uniform solid sphere, the gravitational force inside the sphere varies linearly with distance from the center (i.e. behaves like a harmonic oscillator), while outside the sphere the gravitational field is governed by an inverse square law. This idea is important when determining the net gravitational forces acting on the particle as it moves from regions influenced by the sphere to regions influenced by the shell.

Newton's Shell Theorem

This theorem states that a spherically symmetric shell of mass exerts no net gravitational force on a particle located anywhere inside it, and that outside the shell the gravitational effect can be calculated as though all the mass were concentrated at the center. This concept is crucial when analyzing the gravitational influence of the outer shell on the falling particle.

Please give Ace some feedback