Question

Kepler’s third law states that the square of period of revolution T of a planet around the sun is proportional to the cube of average distance r between sun and planet i.e. T² = Kr³, where K is constant. If the masses of sun and planet are M and m respectively and as per Newton’s law of gravitation the force of attraction between them is F=(GMm)/(r²), where G is gravitational constant, the relation between G and K is described as:

• A. \(GMK = 4\pi^2\)
• B. \(K = G\)
• C. \(K = \dfrac{1}{G}\)
• D. GK = 4π²

Hint:

Kepler’s third law can be derived directly using: Gravitational force = Centripetal force Always compare final expressions term by term.

Answer 1

The Correct Option is D

Step 1: For circular motion of a planet: (mv²)/(r) = (GMm)/(r²)
Step 2: Simplifying: v² = (GM)/(r)
Step 3: Time period of revolution: T = (2π r)/(v) ⟹ T² = (4π² r³)/(GM)
Step 4: Comparing with Kepler’s law T² = Kr³: K = (4π²)/(GM) ⟹ GK = 4π²