Question

If the law of gravitation, instead of being inverse-square law, becomes an inverse-cube law- ______.

1. planets will not have elliptic orbits.
2. circular orbits of planets is not possible.
3. projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic.
4. there will be no gravitational force inside a spherical shell of uniform density.

Answer 1

a and 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 the radius of orbiting planet)

⇒ v = orbital speed = `sqrt(GM)/a`

⇒ `v ∝ 1/a`

Time period of revolution of a planet T = `(2πa)/v = (2πa)/(sqrt(GM)/a) = (2πa^2)/sqrt(GM)`

⇒ `T^2 ∝ a^4`  

Hence, the orbit will not be elliptical.   .....[For elliptical orbit T² ∝ a³]

As force F = `((GM)/a^3)m = g m`

Where, `g^' = (GM)/a^3`

As g', acceleration due to gravity is constant, hence path followed by a projectile will be approximately parabolic. (as T ∝ a²)