Orbital Motion (Edexcel A Level Physics): Revision Note

Orbital Motion

Newton's Law of Gravitation & Orbits

• 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

  • This centripetal force is perpendicular to the velocity 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 from Newton's Law of Gravitation to the centripetal force for a planet or satellite in orbit gives:

$F_G=F_{centripetal}$

$\frac{GMm}{r^2}=\frac{m{v}^2}{r}$

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

$v^2=\frac{GM}{r}$

• Where:

  • v = linear speed of the mass in orbit (m s^–1)

  • G = Newton's Gravitational Constant (N m² kg^–2)

  • M = mass of the object being orbited (kg)

  • r = orbital radius (m)

• This means that all satellites, whatever their mass, will travel at the same speed v in a particular orbital radius r

Newton's Laws of Motion & Orbits

• Newton's first law of motion states that a body remain at rest or at constant velocity unless a resultant force acts on it

• Bodies in orbit do have a resultant force acting on them

  • This is the gravitational force due to the mass M being orbited

  • This force is a centripetal force because it acts towards the centre of M, perpendicular to the velocity of the planet or satellite

• Therefore, since the direction of a planet or satellite orbiting in circular motion is constantly changing, it must be accelerating

  • This is called centripetal acceleration

A satellite in orbit around the Earth travels in circular motion

Time Period & Orbital Radius Relation

• A planet or a satellite orbits in circular motion

  • Therefore, its orbital time period T is the time taken to travel the circumference of the orbit 2πr

• This means the linear speed, or orbital speed v is:

$v=\frac{2\mathrm{\pi }r}{T}$

• This is a result of the well-known equation speed = distance / time

• Equating the two equations for orbital speed gives:

$v^2={\left(\frac{2\mathrm{\pi }r}{T}\right)}^2=\frac{GM}{r}$

• Squaring out the brackets and rearranging for T² gives the equation relating the time period T and orbital radius r:

$T^2=\frac{4{\mathrm{\pi }}^2{r}^3}{GM}$

• Where:

  • T = time period of the orbit (s)

  • r = orbital radius (m)

  • G = Newton's Gravitational Constant (N m² kg^–2)

  • M = mass of the object being orbited (kg)

• The equation shows the relationship between the orbital period and the orbital radius for any planet or satellite in orbit

• It is summarised mathematically as:

$T^2\propto r^3$