An infinite number of spheres,each of mass m, | vedclass

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

Question

(B) The gravitational field $E$ at the origin due to a point mass $m$ at distance $r$ is given by $E = \frac{Gm}{r^2}$.Since the masses are placed at

Vedclass · Accepted Answer

$\frac{4}{3} Gm$

Vedclass · Answer

(B) The gravitational field $E$ at the origin due to a point mass $m$ at distance $r$ is given by $E = \frac{Gm}{r^2}$.Since the masses are placed at distances $r = 1, 2, 4, 8, 16, \dots$,the total gravitational field at the origin is the sum of the fields due to each mass:$E = \frac{Gm}{1^2} + \frac{Gm}{2^2} + \frac{Gm}{4^2} + \frac{Gm}{8^2} + \dots$$E = Gm \left( 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \dots \right)$The term in the bracket is an infinite geometric progression with the first term $a = 1$ and common ratio $r = \frac{1}{4}$.The sum of an infinite geometric series is $S = \frac{a}{1 - r}$.$S = \frac{1}{1 - 1/4} = \frac{1}{3/4} = \frac{4}{3}$.Therefore,the total gravitational field is $E = Gm \left( \frac{4}{3} \right) = \frac{4}{3} Gm$.

An infinite number of spheres,each of mass $m$,are placed on the $X$-axis at distances $1, 2, 4, 8, 16, \dots$ meters from the origin. The magnitude of the gravitational field at the origin is

Similar Questions

$3$ particles each of mass $m$ are kept at the vertices of an equilateral triangle of side $L$. The gravitational field at the centre due to these particles is

Explain gravitational intensity.

In a gravitational field,the gravitational potential is given by $V = -\frac{K}{x} \ (J/kg)$. The gravitational field intensity at point $(2, 0, 3) \ m$ is

If a cavity is made inside a solid sphere,then the gravitational field inside the cavity is

The mass density of a planet of radius $R$ varies with the distance $r$ from its centre as $\rho(r) = \rho_{0} \left(1 - \frac{r^{2}}{R^{2}}\right)$. Then the gravitational field is maximum at:

Vedclass Test Series

Exam Paper Generator

Online Exam Module