# Kepler’s Third Law equation derivation-Time period of Satellite revolution

We will derive the equation for Kepler’s Third Law using the concept of Period of Revolution and the equation of orbital velocity. In this process, the equation of the Time Period Of Revolution of the earth satellite would be derived as well. We’ll also solve sample numerical problems here using this law. So let’s start!

## Kepler’s Third Law

Kepler’s Third Law or 3rd Law of Kepler is an important Law of Physics, which talks about the period of its revolution and how the period of revolution of a satellite depends on the radius of its orbit.

This law states that the square of the Orbital Period of Revolution is directly proportional to the cube of the radius of the orbit.

## Time Period of Revolution of the Satellite about the earth and Kepler’s Third Law derivation

Say, the Period of revolution is T and the radius of the orbit is r.

And r = R + h = Radius of the earth + height of the satellite from the surface of the earth.

### Time Period of revolution of earth satellite

So the period of revolution = T = (2 π r) /V, where V is the Orbital Velocity of the satellite. ……………………….. (1)

Now from the previous post on Orbital Velocity of Satellite, we know that orbital velocity can be expressed as: V = R √(g/r) = R (g/r)^(1/2).

Here g is the acceleration due to gravity and R is the radius of the earth …………….(2)

From equation 1 and 2, we get,

T = (2 π r)/V = [(2 π r)] /[R (g/r)(1/2)]
=[ 2π / R]. √[r3/g]

Here we get an important equation for the Period of Revolution of earth satellite:

T = [ 2π / R]. √[r3/g] ……………………………(3) where R is the radius of the earth and r = R + h

In the next section, we will use the above equation to derive Kepler’s Third Law

### Derivation of Kepler’s Third Law

Now if we square both sides of equation 3 we get the following:

T^2 =[ (4 . π2)/(R2)]. r3/g ……………………………(4)

Here, (4. π2)/(R2) and g are constant as the values of π (Pi), g, and R are not changing with time.

So we can say, T2 ∝ r3. …………….. (5) .[Kepler’s Third Law equation]

Here this Kepler’s Third Law equation says that the square of the Orbital Period of Revolution is directly proportional to the cube of the radius of the orbit.

If the orbit is not circular in the truest sense and rather elliptical, then this law states like this,
the square of the Orbital Period of Revolution varies with the cube of the semi-major axis of the orbit.

This is known as Kepler’s Third Law.

From this equation of Kepler’s Third Law, it comes out clearly that the mass of the object in the revolution has no effect on the Period of Revolution.

And this law is applicable for the revolution of any planet and satellite.

Now let’s solve a numerical problem using this formula, in the next paragraph.

### Kepler’s Third Law – Sample Numerical Problem using Kepler’s 3rd law:

Two satellites Y and Z are rotating around a planet in a circular orbit. The radii of the orbits for Y and Z are 4R and R respectively. Now if the speed of Y satellite is 3v then what is the speed of satellite Z?

Solution:

We would use Kepler’s third law here.

Say, the radius of the orbit of Y and Z are Ry and Rz respectively.
As given data, Ry = 4R and Rz = R

From the 3rd Law of Kepler,
Ty^2/Tz^2 =Ry^3/Rz^3 = (4R)^3/R^3 = 64.  So, Ty/Tz = 8 ……………. (1)

Continuing with this equation (equation 1) and bringing in the orbital velocity in it we get:

(2 π Ry/Vy) / (2 π Rz/Vz) = 8

=> (Ry Vz) /(Rz Vy) = 8

=> (Ry/Rz).(Vz/Vy) = 8

=> (4R/R) .(Vz/Vy) = 8

=> (Vz/Vy) = 8/4 = 2

=> Vz = 2. Vy = 2. 3v = 6v

So the speed of satellite Z is 6v (answer)

### Period of revolution of earth satellite – Numerical Problem

Q1) What is the value of the Period of Revolution of the earth satellite? ignore the height of the satellite above the surface of the earth.

Given: gravitational acceleration (g) = 10 m/s^2. Radius of earth = R = 6400 km. Take π as 3.14

Solution:

Period of revolution = T = [ 2π / R]. √[r ^3/g]
Here r = R + h. As h is to be ignored according to the question, so r =R here.

So T = 2 π √(R/g) ……………. (1)

Given: R= 6400 km = 6400 X 10^3 meter. g = 10 m/s^2
Putting the values given, we get,

T = 2 X 3.14 X √ [(6400X10^3)/10] = 2 X 3.14 X 800 = 5024 seconds = 83.73 minutes.

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