Question

Kepler's third law states that the square of the period of revolution (T) of a planet around the sun is proportional to the third power of average distance r between sun and planet i.e. T² = Kr³, where K is a 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 the 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 is a direct consequence of Newton’s law of gravitation.

Answer 1

The Correct Option is A

Step 1: For circular motion, (GMm)/(r²) = (mv²)/(r) ⟹ v² = (GM)/(r)
Step 2: Time period: T = (2π r)/(v) ⟹ T² = (4π² r³)/(GM)
Step 3: Comparing with T² = Kr³: K = (4π²)/(GM) ⟹ GMK = 4π²