Edexcel A Level Physics:復(fù)習(xí)筆記12.7 Orbital Motion
Orbital Motion
Newton's Law of Gravitation & Orbits
- Since most planets and satellites have a near circular orbit, the gravitational force?FG?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:
FG?=?Fcentripetal
- The mass of the satellite?m?will cancel out on both sides to give:
- 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:
- This is a result of the well-known equation speed = distance / time
- Equating the two equations for orbital speed gives:
- Squaring out the brackets and rearranging for?T2?gives the equation relating the time period?T?and orbital radius?r:
- 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:
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?has a radius of?R2. Both have linear speed?v?and an angular speed ? about?B.
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 ?
Exam Tip
This worked example helps you practise two crucial techniques that are often examined:
- The centripetal force is expressed as
or equivalently as
. In our case the angular speed is given in the question, so it is best to use the latter expressionwhen equating the centripetal force to the gravitational force. - The gravitational force is given as
but note that the distance r in this question is given as a sum, 
. You should remember that r is defined as the distance between the centre of masses of M and m, therefore,
and so
.
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