Period of Satellite's Orbital Motion Around Earth

[Asla]

Homework Statement

An artificial satellite is traveling in a circular orbit around the earth.The radius of the orbit from the Earth's center is r.Assume that the Earth is a uniform sphere of radius R.The magnitude of acceleration due to gravity at the surface is g.What is the period of the satellites orbital motion?
a)2pi r/sqrt(gR)
b)2pi R/sqrt(gr)
c)2pi r/R sqrt(r/g)
d)2pi R/r sqrt(R/g)

I am thinking that the problem is somehow related to uniform circular motion but I hit my first problem while trying to get the centripetal acceleration that is acting on the satellite because the give gravitational force is supposed to be on the surface of the earth.

 

[pbuk]

I hit my first problem while trying to get the centripetal acceleration that is acting on the satellite because the give gravitational force is supposed to be on the surface of the
earth.

How does the force of gravity change with distance?

So if the force of gravity on the surface of the earth, i.e. at distance R from the centre of the Earth is g, what is the force of gravity at distance r?

 

[Asla]

Yap I think that is the approach.Gravity acting on the satellite is R^2/r^2 (g) through which you get to (c) as the answer

 

[pbuk]

That's it

 

[paranormal]

I would like to clarify that the period of a satellite's orbital motion around Earth is determined by the relationship between its orbital radius and the gravitational force acting on it. The correct answer to the given problem is option b) 2pi R/sqrt(gr).

To understand this, we need to use the formula for centripetal acceleration, which is given by a = v^2/r, where v is the orbital velocity and r is the radius of the orbit. In this case, the gravitational force acting on the satellite is providing the centripetal force needed to maintain the circular motion, and can be expressed as F = GmM/r^2, where G is the universal gravitational constant, m is the mass of the satellite, and M is the mass of Earth.

Equating these two equations, we get v^2/r = GmM/r^2, which can be rearranged to get v = sqrt(GM/r). Now, we know that the period of a circular motion is given by T = 2pi r/v. Substituting v in this equation, we get T = 2pi r/sqrt(GM/r). Since we are dealing with Earth's gravitational force, we can replace M with the mass of Earth, which is given by ME = (4/3)pi R^3p, where R is the radius of Earth and p is the density. Substituting this in the equation for T, we get T = 2pi r/sqrt(G(4/3)pi R^3p/r). Simplifying this further, we get T = 2pi R/sqrt(gr), which is the correct answer.

In conclusion, the period of a satellite's orbital motion around Earth is determined by the relationship between its orbital radius and the gravitational force acting on it, and can be calculated using the formula T = 2pi R/sqrt(gr).