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

Options

A. planets will not have elliptic obits
B. circular orbits of planets is 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 becomes an inverse cube law, then we can write, for a planet of mass `m` revolving around the sun of mass M,
`F = (GMm)/(a^(3)) = (mv^(2))/(a)` (where a is radius of orbiting planet)
`rArr` v = orbital speed `= (sqrt(GM))/(a) rArr v prop (1)/(a)`
Time period of revolution of a planet `T = (2 pia)/(v) = (2 pi a)/((sqrt(GM))/(a)) = (2 pi a^(2))/(sqrt(GM))`
`rArr T^(2) prop a^(4)`
Hence, orbit will not be elliptical [for elliptical orbit `T^(2) prop a^(3)`]
As force `F = ((GM)/(a^(3))) m = g' m`
where, `g' = (GM)/(a^(3))`
As g', acceration due to gravity is constant, hence path followed by a projectile will be approximately parabolic. (as `T prop a^(2)`)