Question

the planet move around the sun in nearly circular orbit assuming that the period of rotation t depends upon the radius of the orbit the mass of the sun and gravitational constant show that the square of the time period is directly proportional to the cube of the orbit radius

Answer 1

YOUR ANSWER GOES IN THE WAY OF LAW STATED BY KEPLER.

3rd LAW OF KEPLER:

T² ∝R³

It is known as Law of periods..

Let us consider a planet P of mass m moving with a velocity v around the sun of mass M in a circular orbit of radius r.

The gravitational force of attraction of the sun on the planet is,

F=GMm/r².

The centripetal force is,F = mv²/r.

equating the two forces,

mv²/r=GMm/r².

v²=GM/r -----›(i)

If T be the period of revolution of the planet around the sun, then

v=2πr/T-------›(ii)

Substituting (ii) in (i)

4π²r²/T²=GM/r

r³/T²=GM/4π²

GM is a constant for any planet.

•°• T²∝R³.

THANK YOU!!☺️

ALL THE BEST.

Answer 2

Hey!!mate

YOUR ANSWER GOES IN THE WAY OF LAW STATED BY KEPLER.

3rd LAW OF KEPLER:

T² ∝R³

It is known as Law of periods..

Let us consider a planet P of mass m moving with a velocity v around the sun of mass M in a circular orbit of radius r.

The gravitational force of attraction of the sun on the planet is,

F=GMm/r².

The centripetal force is,F = mv²/r.

equating the two forces,

mv²/r=GMm/r².

v²=GM/r -----›(i)

If T be the period of revolution of the planet around the sun, then

v=2πr/T-------›(ii)

Substituting (ii) in (i)

4π²r²/T²=GM/r

r³/T²=GM/4π²

GM is a constant for any planet.

•°• T²∝R³.


hope this helps....