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Question
If the law of gravitation, instead of being inverse-square law, becomes an inverse-cube law- ______.
- planets will not have elliptic orbits.
- circular orbits of planets is not possible.
- projectile motion of a stone thrown by hand on the surface of the earth will be approximately parabolic.
- there will be no gravitational force inside a spherical shell of uniform density.
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Solution
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)
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