## Period For a GPS Satellite

## The satellites in the Global Positioning System (GPS) orbit at an altitude of 21400 km from earth.

What is the period of the satellites?

**46100 s**## Approach

The period of the satellites is the distance traveled for 1 revolution divided by the speed of the satellites. The distance for one revolution is 2πr while the speed can be obtained using the Law of Gravitation and Newton's Second Law of motion.

## Solution

The Law of Gravitation state...

F=G*(m1*m2/r^2)

This force is the centripetal force acting on the satellites and by Newton's Second Law of motion is equal to mass times acceleration.

G*(m1m2/r^2) = (m2*V^2)/r = V= sqrt(Gm1*1/r)

Here m1 is the mass of the Earth and the distance r is measured from the center of the earth i.e., r = (6370 + 21400)*1000 m = 27770000 m. (The radius of the earth is added to the altitude since altitude is distance from the surface of the earth.) Putting the values into the above formula gives v = 3787 m/s. The period T = 2πr/v = 46100 s or 12.8 hr.

Note: The actual GPS satellites have an altitude of 20200 km and a period of 12.0 hrs.

F=G*(m1*m2/r^2)

This force is the centripetal force acting on the satellites and by Newton's Second Law of motion is equal to mass times acceleration.

G*(m1m2/r^2) = (m2*V^2)/r = V= sqrt(Gm1*1/r)

Here m1 is the mass of the Earth and the distance r is measured from the center of the earth i.e., r = (6370 + 21400)*1000 m = 27770000 m. (The radius of the earth is added to the altitude since altitude is distance from the surface of the earth.) Putting the values into the above formula gives v = 3787 m/s. The period T = 2πr/v = 46100 s or 12.8 hr.

Note: The actual GPS satellites have an altitude of 20200 km and a period of 12.0 hrs.