$A$ satellite of mass $m$ is at a distance $a$ from a star of mass $M$. The speed of the satellite is $u$. Suppose the law of universal gravity is $F = -G \frac{Mm}{r^{2.1}}$ instead of $F = -G \frac{Mm}{r^2}$. Find the speed of the satellite when it is at a distance $b$ from the star.

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.

Out of the following,the only incorrect statement about satellites is

Which of the following astronomers first proposed that the Sun is static and the Earth revolves around the Sun?

Match List-$I$ with List-$II$:

$(a)$ Gravitational constant $(G)$	$(i)$ $[L^{2}T^{-2}]$
$(b)$ Gravitational potential energy	$(ii)$ $[M^{-1}L^{3}T^{-2}]$
$(c)$ Gravitational potential	$(iii)$ $[LT^{-2}]$
$(d)$ Gravitational intensity	$(iv)$ $[ML^{2}T^{-2}]$

Choose the correct answer from the options given below:

Two bodies,each of mass $M$,are kept fixed with a separation $2L$. $A$ particle of mass $m$ is projected from the mid-point of the line joining their centres,perpendicular to the line. The gravitational constant is $G$. The correct statement$(s)$ is/are:
$(a)$ The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $4 \sqrt{\frac{GM}{L}}$
$(b)$ The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $2 \sqrt{\frac{GM}{L}}$
$(c)$ The minimum initial velocity of the mass $m$ to escape the gravitational field of the two bodies is $\sqrt{\frac{2GM}{L}}$
$(d)$ The energy of the mass $m$ remains constant