CIE A Level Physics复习笔记13.1.3 Circular Orbits in Gravitational Fields

• Since most planets and satellites have a near circular orbit, the gravitational force F_G between the sun or another planet provides the centripetal force needed to stay in an orbit
• Both the gravitational force and centripetal force are perpendicular to the direction of travel of the planet
• Consider a satellite with mass m orbiting Earth with mass M at a distance r from the centre travelling with linear speed v

• Equating the gravitational force to the centripetal force for a planet or satellite in orbit gives:

• The mass of the satellite m will cancel out on both sides to give:

• This means that all satellites, whatever their mass, will travel at the same speed v in a particular orbit radius r
• Recall that since the direction of a planet orbiting in circular motion is constantly changing, it has centripetal acceleration

A satellite in orbit around the Earth travels in circular motion

Kepler’s Third Law of Planetary Motion

• For the orbital time period T to travel the circumference of the orbit 2πr, the linear speed v can be written as

• This is a result of the well-known equation, speed = distance / time
• Substituting the value of the linear speed v into the above equation:

• Rearranging leads to Kepler’s third law equation:

• The equation shows that the orbital period T is related to the radius r of the orbit. This is known as Kepler’s third law:

For planets or satellites in a circular orbit about the same central body, the square of the time period is proportional to the cube of the radius of the orbit

• Kepler’s third law can be summarised as:

Maths Tip

• The ∝ symbol means ‘proportional to’
• Find out more about proportional relationships between two variables in the “proportional relationships ” section of the A Level Maths revision notes

(1) The angular speed of ⍵ of M₁

Step 1: Equating the centripetal force of mass M₁ to the gravitational force between M₁ and M₂

Step 2: M₁ cancels on both sides

Step 3: Rearrange for angular velocity ⍵

Step 4: Square root both sides

(2)  The time period T for each star in terms of angular speed ⍵

Step 1: Angular speed equation with time period T

Step 2: Rearrange for T

Step 3: Substitute in ⍵