Question

Kepler's third law states that square of period of revolution (T) of a planet around the sun, is proportional to third power of average distance r between sun and planet i.e. $T^2 = Kr^3$ here K is constant. If the masses of sun and planet are M and m respectively then as per Newton's law of gravitation force of attraction between them is $F = \frac {GMm}{r^2}$, here G is gravitational constant. The relation between G and K is described as

• A. $K = G$
• B. $K = \frac {1}{G}$
• C. $GK = 4 \pi^2 $
• D. $GMK = 4 \pi^2$

Answer 1

The Correct Option is D

To find the relation between the gravitational constant \( G \) and the proportionality constant \( K \) from Kepler's third law, we must connect the two fundamental equations: Newton's law of gravitation and Kepler's third law.

1. Kepler's Third Law: According to Kepler's third law, the square of the period of revolution \( T \) of a planet around the sun is proportional to the cube of the average distance \( r \) between the sun and the planet:

T^2 = Kr^3

2. Newton's Law of Gravitation: The gravitational force \( F \) between two masses \( M \) (sun) and \( m \) (planet) is given by:

F = \frac{GMm}{r^2}

3. Centripetal Force and Orbital Motion: For a planet in circular motion around the sun, the gravitational force provides the necessary centripetal force.

Centripetal force \( F \) is also related to the motion of the planet by: F = \frac{mv^2}{r}, where \( v \) is the orbital velocity of the planet.

4. Equate gravitational force to centripetal force:

\frac{mv^2}{r} = \frac{GMm}{r^2}

This simplifies to: v^2 = \frac{GM}{r}

5. Relate \( v \) and \( T \): Using \( v = \frac{2\pi r}{T} \) for circular orbits:

\left(\frac{2\pi r}{T}\right)^2 = \frac{GM}{r}

6. Simplify the equation:

\frac{4\pi^2 r^2}{T^2} = \frac{GM}{r}

Rearrange to find \( T^2 \):

T^2 = \frac{4\pi^2 r^3}{GM}

7. Comparing with Kepler's law \( T^2 = Kr^3 \) gives us:

K = \frac{4\pi^2}{GM}

Therefore, the correct relationship between \( G \), \( K \), and \( M \) is:

Correct Answer: GMK = 4\pi^2

This concludes the connection between Kepler's third law as represented by the constant \( K \), and the gravitational parameters described by \( G \) and \( M \).