If a person with a spring balance and a body hanging from it goes up and up in an aeroplane,then the reading of the weight of the body as indicated by the spring balance will

$A$ particle of mass $m$ starts from rest at a distance $R$ from the centre and along the axis of a fixed ring of radius $R$ and mass $M$. Its velocity at the centre of the ring is:

Fill in the blanks:
$(a)$ In the region inside the Earth,the acceleration due to gravity is proportional to the distance from the center of the Earth as ..... .
$(b)$ If the Earth shrinks such that its radius becomes half and its mass remains constant,the weight of an object on the Earth increases by a factor of ......... .
$(c)$ The orbital velocity of a geostationary satellite of the Earth is approximately ............ .

$A$ uniform sphere $A$ with radius $R$ exerts a gravitational force $F$ on a small particle $B$ situated at a distance $2R$ from the centre of the sphere. $A$ spherical portion of diameter $R$ is cut from the sphere $A$ such that its centre lies on the line connecting the centre of sphere $A$ and particle $B$. If $F^{\prime}$ is the new gravitational force between the remaining part of the sphere $A$ and the particle $B$, then find the correct relation between $F$ and $F^{\prime}$.

Three point masses,each of mass $m$,are kept at the corners of an equilateral triangle of side $L$. The system rotates about the center of the triangle without any change in the separation of masses during rotation. The period of rotation is directly proportional to

During the motion of a planet from perihelion to aphelion,the work done by the gravitational force of the sun on it is ...........