An infinite number of objects,each of mass 1 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) The gravitational potential $V$ at a point due to a mass $m$ at distance $r$ is given by $V = -\frac{Gm}{r}$.Here,$m = 1 \ kg$. The objects are pl

Vedclass · Accepted Answer

$4G$

Vedclass · Answer

(D) The gravitational potential $V$ at a point due to a mass $m$ at distance $r$ is given by $V = -\frac{Gm}{r}$.Here,$m = 1 \ kg$. The objects are placed at $x = \pm 1, \pm 2, \pm 4, \pm 8, \ldots \ m$.Since there are two objects at each distance $r$ (one at $+r$ and one at $-r$),the total potential $V_{total}$ at $x=0$ is:$V_{total} = \sum -\frac{Gm}{r_i} = -G(1) \left[ \frac{1}{1} + \frac{1}{1} + \frac{1}{2} + \frac{1}{2} + \frac{1}{4} + \frac{1}{4} + \ldots ight]$$V_{total} = -G \left[ 2 \left( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots ight) ight]$The term in the bracket is a geometric progression with first term $a=1$ and common ratio $r=1/2$. The sum is $S = \frac{a}{1-r} = \frac{1}{1 - 1/2} = 2$.Therefore,$V_{total} = -G 	imes 2 	imes 2 = -4G$.The magnitude of the potential is $|V_{total}| = 4G$.

Similar Questions

$A$ mass of $50 \, \text{kg}$ is placed at the centre of a uniform spherical shell of mass $100 \, \text{kg}$ and radius $50 \, \text{m}$. If the gravitational potential at a point $25 \, \text{m}$ from the centre is $V \, \text{J/kg}$,find the value of $V$.

$A$ uniform solid sphere of mass $M$ and radius $R$ is surrounded symmetrically by a uniform thin spherical shell of equal mass and radius $2R$. The value of gravitational potential at a distance $\frac{3}{2}R$ from the center is:

$A$ particle of mass $M$ is placed at the centre of a uniform spherical shell of mass $2M$ and radius $R$. The gravitational potential on the surface of the shell is

$A$ particle of mass $m$ is released from infinity and it moves towards a large solid sphere of mass $M$ and radius $R$ made of a jelly-like material. Neglecting the resistance offered by the material of the sphere,find the velocity of the particle when it crosses the centre of the sphere.

If the potential energy of a body of mass $m$ on the surface of the Earth is taken as zero,then its potential energy at height $h$ above the surface of the Earth is ($R$ is the radius of the Earth and $M$ is the mass of the Earth).

Vedclass Test Series

Exam Paper Generator

Online Exam Module