Question

The slope of the graph plotted between the square of the time period of a planet \( T^2 \) and the cube of its mean distance from the sun \( r^3 \) is:

• A. \( \frac{4\pi^2}{GM} \)
• B. \( 4\pi GM \)
• C. \( \frac{4\pi G}{M} \)
• D. \( \frac{4\pi^2 M}{G} \)
• E. Zero

Hint:

For planetary motion, Kepler's third law provides a direct relationship between the square of the time period and the cube of the mean distance.

Answer 1

The Correct Option is A

Step 1: Kepler’s third law states: \[ T^2 = \frac{4\pi^2 r^3}{GM} \] where: - \( T \) is the orbital period of the planet,
- \( r \) is the mean distance from the sun,
- \( G \) is the gravitational constant,
- \( M \) is the mass of the sun.
Step 2: The slope of the graph between \( T^2 \) and \( r^3 \) will be: \[ {slope} = \frac{4\pi^2}{GM} \] Step 3: This is the required slope, and it depends on the mass of the sun and the gravitational constant.