A very long (length L) cylindrical galaxy is | vedclass

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

Question

(A) For a very long cylindrical mass distribution,the gravitational field $E$ at a distance $r$ $(r > R)$ from the axis is given by Gauss's Law for gr

Vedclass · Accepted Answer

$T \propto r$

Vedclass · Answer

(A) For a very long cylindrical mass distribution,the gravitational field $E$ at a distance $r$ $(r > R)$ from the axis is given by Gauss's Law for gravitation: $E = \frac{2GM}{Lr}$.The gravitational force acting on a star of mass $m$ is $F = mE = \frac{2GMm}{Lr}$.Since the star is orbiting in a circular path,this gravitational force provides the necessary centripetal force: $\frac{mv^2}{r} = \frac{2GMm}{Lr}$.Substituting $v = r\omega = r(\frac{2\pi}{T})$,we get: $m r (\frac{2\pi}{T})^2 = \frac{2GMm}{Lr}$.Simplifying the equation: $r \frac{4\pi^2}{T^2} = \frac{2GM}{Lr}$.Rearranging for $T^2$: $T^2 = \frac{4\pi^2 L}{2GM} r^2$.Therefore,$T^2 \propto r^2$,which implies $T \propto r$.

Similar Questions

The ratio of the $K.E.$ required to be given to a satellite to escape Earth's gravitational field to the $K.E.$ required to be given so that the satellite moves in a circular orbit just above Earth's atmosphere 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

The magnitudes of the gravitational field at distances $r_1$ and $r_2$ from the centre of a uniform sphere of radius $R$ and mass $M$ are $F_1$ and $F_2$ respectively. Then-

$A$ spacecraft travels in a straight line from the Earth to the Moon. How will its weight change during this journey?

Two identical thin uniform rods of mass $M$ and length $L$ are placed in a line at a separation of $L$. Find the gravitational force acting between them.

Vedclass Test Series

Exam Paper Generator

Online Exam Module