Circular Orbits in Gravitational Fields (OCR A Level Physics)

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Centripetal Force on a Planet

  • Typically, most planets and satellites have a near circular orbit
    • Therefore, the gravitational force F between the Sun and another planet provides the centripetal force needed to stay in an orbit
  • Since the gravitational force is centripetal, it is perpendicular to the direction of travel of the planet

  • Consider a satellite with mass m orbiting the Earth, with mass M, at a distance r from its centre and travelling with linear speed v
  • The gravitational force F on the satellite is centripetal, therefore:

F = Fcentripetal

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

Circular Orbits Equation 1

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

Circular Orbits Equation 2

  • This can be reduced to give an equation for the orbital speed vas:

v equals square root of fraction numerator G M over denominator r end fraction end root

  • Where:
    • v = orbital speed of the mass in orbit (m s-1)
    • G = Newton's Gravitational Constant
    • 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 orbit radius r
    • Since the direction of a planet orbiting in circular motion is constantly changing, it has centripetal acceleration

Circular motion satellite, downloadable AS & A Level Physics revision notes

A satellite in orbit around the Earth travels in circular motion

Circular Orbits in Gravitational Fields

  • Assuming a planet or a satellite is travelling in circular motion when in orbit, its orbital speed v is given by the distance travelled divided by the orbital period T:

v equals fraction numerator 2 straight pi r over denominator T end fraction

  • This is a result of the well-known equation, speed = distance / time
    • Where the distance is equal to the circumference of a circle = 2πr
  • Recall, orbital speed v is also given by: 

v squared equals fraction numerator G M over denominator r end fraction

  • Therefore:

v squared equals open parentheses fraction numerator 2 straight pi r over denominator T end fraction close parentheses squared equals fraction numerator G M over denominator r end fraction

  • Expanding the brackets gives:

fraction numerator 4 straight pi squared r squared over denominator T squared end fraction equals fraction numerator G M over denominator r end fraction

  • Rearranging for T2 gives the mathematical expression of Kepler's Third Law which relates the time period T and orbital radius as:

T squared equals fraction numerator 4 straight pi squared r cubed over denominator G M end fraction

  • Where:
    • T = time period of the orbit, or "orbital period" (s)
    • r = orbital radius (m)
    • G = Newton's Gravitational Constant
    • M = mass of the object being orbited (kg)

Worked example

A binary star system constant of two stars orbiting about a fixed point B.

The star of mass M1 has a circular orbit of radius R1 and mass M2 also has a circular orbit of radius of R2. Both have linear

speed v and an angular speed ⍵ about B.

Worked example - circular orbits in g fields, downloadable AS & A Level Physics revision notes

State the following formula, in terms of G, M2, R1 and R2

(i) The angular speed ⍵ of M1

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

Circular Orbits Worked Example Part 2

Exam Tip

Many of the calculations in planetary motion questions depend on the equations for circular motion. Be sure to revisit these and understand how to use them! You will be expected to remember the derivation for T squared equals fraction numerator 4 straight pi squared over denominator G M end fraction r cubed , so make sure you understand each step, especially equating the gravitational force and the centripetal force! 

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Katie M

Author: Katie M

Katie has always been passionate about the sciences, and completed a degree in Astrophysics at Sheffield University. She decided that she wanted to inspire other young people, so moved to Bristol to complete a PGCE in Secondary Science. She particularly loves creating fun and absorbing materials to help students achieve their exam potential.

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