Gravitational and electric fields

[confusedpilot]

Homework Statement

By considering the centripetal force which acts on a planet in a circular orbit, show that T^2 is proportional to R~^3 where T is the time taken for one orbit around the sun and R is the radius of the orbit

Homework Equations

I'm thinking keplers third law?

The Attempt at a Solution

C=2piR

P=2piR/ sqrt*GMm/r

rearranged to give P = 2pi sqrt*R^3/GM

Lost now..

 

[Donaldos]

P=2piR/ sqrt*GMm/r

Where did that expression come from?


Find an expression for the radial acceleration in terms of the orbital speed and R.

Then use Newton's second law.

 

[tomcue]

I would like to clarify that the equations and concepts mentioned in the question are correct and relevant. The attempt at a solution is also on the right track. Kepler's third law, also known as the law of harmonies, states that the square of the orbital period (T^2) of a planet is proportional to the cube of its semi-major axis (R^3). This law applies to any two bodies in orbit around each other, including planets and the sun.

The equation P = 2pi sqrt*R^3/GM shows the relationship between the orbital period and the radius of the orbit. By considering the centripetal force acting on a planet in a circular orbit, we can see that the force is equal to the gravitational force between the planet and the sun (F=GMm/r^2). Setting these two forces equal to each other and solving for the orbital period (T), we get T = 2piR/ sqrt*GMm/r. Substituting this into the equation for Kepler's third law, we get T^2 = 4pi^2R^3/GM. This shows that T^2 is indeed proportional to R^3, as stated in the question.

In conclusion, by considering the centripetal force and applying Kepler's third law, we can show that T^2 is proportional to R^3. This relationship is important in understanding the dynamics of orbits and is a fundamental concept in the study of gravitational and electric fields.