A large spherical mass M is fixed at one posi | vedclass

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

Question

(A) Both point masses are connected by a rigid massless rod,so they must have the same acceleration $a$ towards the large mass $M$.Let $F_1$ be the gr

Vedclass · Accepted Answer

$7$

Vedclass · Answer

(A) Both point masses are connected by a rigid massless rod,so they must have the same acceleration $a$ towards the large mass $M$.Let $F_1$ be the gravitational force on the nearer mass $m$ due to $M$,and $F_2$ be the gravitational force on the farther mass $m$ due to $M$.Let $F$ be the tension in the rod. Since the tension is zero,$F = 0$.For the nearer mass $m$:$F_1 = ma \implies \frac{GMm}{(3\ell)^2} = ma \implies a = \frac{GM}{9\ell^2} \quad (i)$For the farther mass $m$:$F_2 = ma \implies \frac{GMm}{(4\ell)^2} = ma \implies a = \frac{GM}{16\ell^2} \quad (ii)$However,the rod also exerts a gravitational force between the two masses $m$. Let this force be $F_g = \frac{Gm^2}{\ell^2}$.For the nearer mass: $F_1 - F_g = ma$For the farther mass: $F_2 + F_g = ma$Equating the two expressions for $ma$:$F_1 - F_g = F_2 + F_g \implies F_1 - F_2 = 2F_g$$\frac{GMm}{9\ell^2} - \frac{GMm}{16\ell^2} = 2 \left( \frac{Gm^2}{\ell^2} \right)$$GMm \left( \frac{16 - 9}{144\ell^2} \right) = \frac{2Gm^2}{\ell^2}$$\frac{7GMm}{144} = 2Gm^2 \implies \frac{7M}{144} = 2m \implies m = \frac{7M}{288}$Comparing with $m = k \left( \frac{M}{288} \right)$,we get $k = 7$.

Similar Questions

Two satellites of masses $m$ and $1.5 m$ are revolving around the earth in two circular orbits of heights $R_E$ and $2 R_E$ respectively,where $R_E$ is the radius of the earth. The ratio of the minimum and maximum gravitational forces exerted by the two satellites on the earth is

$A$ particle of mass $m_0$ is projected from the mid-point of the line joining two fixed particles each of mass $m.$ If the distance of separation between the fixed particles is $l,$ the minimum velocity of projection of the particle so as to escape to infinity is equal to

What is the gravitational force exerted by the sphere of mass $M$ and radius $a$ on a ring of mass $m$ and radius $a$,where the distance between the center of the ring and the center of the sphere is $r = 1.73a = \sqrt{3}a$?

Which one of the following plots represents the variation of gravitational field on a particle with distance $r$ due to a thin spherical shell of radius $R$? ($r$ is measured from the centre of the spherical shell)

Vedclass Test Series

Exam Paper Generator

Online Exam Module