Question

A sphere of density $ρ$ and radius $a$ has a concentric cavity of radius $b$ , as shown in the figure.

(a) Sketch the gravitational force $F$ exerted by the sphere on the particle of mass $m$ , located at a distance $r$ from the centre of the sphere as a function of r in the range $0 \leq r \leq \infty .$

(b) Sketch the corresponding curve for the potential energy $u (r)$ of the system.

Accepted Answer

(a) For $0 < r < b$ , there is no mass inside the spherical shell, i.e., $M = 0$ , so the gravitational force $F = 0$ .

For $b < r < a$ , the mass of the sphere for radius $r$ will be,

$M = \frac{4}{3} π (r^{3} - b^{3}) ρ$

The gravitational force between sphere and particle will be,

$F = \frac{G M m}{r^{2}} = \frac{4}{3} π G ρ m (r - \frac{b^{3}}{r^{2}})$

$F \propto (r - \frac{b^{3}}{r^{2}})$

For $r > a$ , the mass of the shell will be,

$M = \frac{4}{3} π (a^{3} - b^{3}) ρ$

The Gravitational force will be,

$F = \frac{G M m}{r^{2}} = \frac{\frac{4}{3} π (a^{3} - b^{3}) ρ m}{r^{2}}$

$F \propto \frac{1}{r^{2}}$

Therefore, the curve between $F$ and $r$ will be given as,

(b) The potential energy is given by,

$U_{f} - U_{i} = \int_{r_{i}}^{r_{f}} F d r$

For $a < r < \infty$ , we have,

$U (r) - U (\infty) = \int_{\infty}^{r} \frac{\frac{4}{3} π (a^{3} - b^{3}) m ρ}{r^{2}} d r$

$U (r) - 0 = \frac{4}{3} π (a^{3} - b^{3}) ρ {[- \frac{1}{r}]}_{\infty}^{r} = - \frac{4}{3 r} π G m (a^{3} - b^{3}) ρ$

For $b < r < a$ ,

$U (r) - U (a) = \int_{a}^{r} \frac{4}{3} π G m ρ (r - \frac{b^{3}}{r^{2}}) d r$

$U (r) + \frac{4}{3 a} π G m (a^{3} - b^{3}) ρ = \frac{4}{3} π G m ρ {[\frac{r^{2}}{2} + \frac{b^{3}}{r}]}_{a}^{r}$

$U (r) = - \frac{2}{3 r} π G ρ m (3 r a^{2} - 2 b^{3} - r^{3})$

For $0 < r < b$ , $F = 0$ , so,

$U (r) - U (b) = 0$

$U (r) = U (b) = - 2 π G ρ m (a^{2} - b^{2})$ .

Hence, the graph will be like,

Embibe Experts Solutions for Chapter: Gravitation, Exercise 4: Exercise - 4

Embibe Experts Physics Solutions for Exercise - Embibe Experts Solutions for Chapter: Gravitation, Exercise 4: Exercise - 4

Questions from Embibe Experts Solutions for Chapter: Gravitation, Exercise 4: Exercise - 4 with Hints & Solutions

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