If the law of gravitational, instead of being inverse-square law, becomes an inverse-cube law

Options

A. planets will not have elliptic orbits.
B. circular orbits of planets are not possible.
C. projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic.
D. there will be no gravitational force inside a spherical shell of uniform density.

Answer

A, C

Explanation

If the law of gravitation is inverse-cube law, then gravitational force is given by
`F = (GMm)/(r^(3))`
This force will now provide the necessary centripetal force so that an object of smaller mass m can revolve around another object of heavier mass M.
`:. (GMm)/(r^(3)) = (mv^(2))/(r )`
`rArr v = (sqrt(GM))/(r )`
Time period of revolution of the planet is
`T = (2pi r)/(v) = (2pi r)/((sqrt(GM))/(r )) = (2pi r^(2))/(sqrt(GM))`
Squaring on both side
`rArr T^(2) = (4pi^(2) r^(4))/(GM)`
`rArr T^(2) prop r^(4)`
This shows that planets will not have elliptical orbits.
Acceleration due to gravity near the surface of the earth can be calculated as below :
`mg = (GMm)/(r^(3)) rArr g = (GM)/(r^(3))`
Now if a projectile is dropped from a certain height distance travellled by the projectile is
`s = u t + (1)/(2) a t^(2)`
`= 0 + (1)/(2) g t^(2) = (GM)/(2r^(3)) t^(2)`
`s prop t^(2)`
Motion of the projectile with respect to time near the surface of earth will be approximately parabolic.
The gravitational force inside a spherical shell of uniform density cannot be zero as the gravitational force varies inversely with the distance r from the centre of the shell.
The correct option are (a) and (c ).