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2.6.2 Orbital Motion

By the end of this lesson you can:

  • Explain how satellites stay in orbit
  • Calculate orbital speed and period
  • Compare circular orbits at different radii
Find your course: NSW Year 11NSW Year 12: Module 5, IQ3QLD Unit 3: Gravity and motionVIC Unit 3: Motion in two dimensionsVIC Unit 3: Fields and motionAP Physics 1: Unit 2AP Physics 1: Unit 6IB: D.1 Gravitational fields
🎥 Watch📝 Practise🕹️ Explore🎯 Exam ready🍎 Teachers
🎥Watch
Learn the idea: a short explanation, then a check.
If the satellite is in orbit around a larger object it generally follows either an elliptical path or a circular path. This could be a case of an artificial satellite around the planet, or a plant around a star.
The reason it stays in orbit is simply because the gravitational force holds it there, and this gravitational force ends up being the centripetal force.
As a result, we are able to establish mathematical relationships between how fast the satellite is going and how high the satellite is, as well as the mass of the central object.
We will start by first examining Newton' Cannon. Newton developed a thought experiment that involved a cannon on top of a mountain. He realised, when fired, it will fall to the ground. But what if the cannon fired fast enough , that the trajectory would have to take into account the curvature of the earth. If he fired it fast enough, the cannon would still fall, but never reach the ground. It would be in essence, in orbit.
Have a go at determining the speed required with this animation

​Now watch this video which examines the physics principles for circular orbits.

📝Practise
Worked examples first, then problems to try, then worksheets.
Sample Problem
We are now ready to try a sample problem
Below is a sample problem with a video that explain how to solve it. It is suggested you try the problem beforehand, as this actually aids understanding, even if you are unsure if you are correct.
Picture

Constants used: G = 6.674 × 10-11 N m2 kg-2, mass of the Earth M = 6.0 × 1024 kg, radius of the Earth R = 6.37 × 106 m.

1. A satellite is in orbit at an altitude of 500 km. What is the velocity of this satellite?

Given: h = 500 km = 5.0 × 105 m

Step 1 (Orbital radius): r = R + h = 6.37 × 106 + 5.0 × 105 = 6.87 × 106 m

Step 2 (Gravity provides the centripetal force): GMm / r2 = mv2 / r → v = √(GM / r)

v = √[(6.674 × 10-11 × 6.0 × 1024) / (6.87 × 106)] = √(5.83 × 107) ≈ 7.63 × 103 m/s

Answer: 7.63 km/s
2. The Moon is 384,000 km away from the Earth, measured from their respective centres. How fast is it travelling?

Given: r = 384,000 km = 3.84 × 108 m

Step 1 (Gravity provides the centripetal force): v = √(GM / r)

Step 2 (Calculate speed): v = √[(6.674 × 10-11 × 6.0 × 1024) / (3.84 × 108)] = √(1.043 × 106) ≈ 1.02 × 103 m/s

Answer: 1.02 km/s
3. The Sun is part of the Milky Way galaxy and rotates about the centre of this galaxy. The Sun's radius of rotation is about 2.7 × 1020 m and its period is about 6.3 × 1015 seconds. If our Sun has a mass of approximately 2 × 1030 kg, calculate the number of similar suns or stars that make up the mass of the Milky Way. (Assume most of the galaxy's mass is located centrally.)

Given: r = 2.7 × 1020 m, T = 6.3 × 1015 s, MSun = 2 × 1030 kg

Step 1 (Mass of the galaxy): Gravity provides the centripetal force, GMm / r2 = m(4π2r / T2) → M = 4π2r3 / (GT2)

M = 4π2(2.7 × 1020)3 / [6.674 × 10-11 × (6.3 × 1015)2] = (7.77 × 1062) / (2.65 × 1021) ≈ 2.93 × 1041 kg

Step 2 (Number of suns): N = (2.93 × 1041) / (2 × 1030) ≈ 1.47 × 1011

Answer: number of 'suns' = 1.47 × 1011
4. NROL-76 is a military satellite launched in 2017 that has one of the lowest orbits. If its period is 90.5 minutes, determine its altitude.

Given: T = 90.5 min = 5430 s

Step 1 (Orbital radius): GMm / r2 = m(4π2r / T2) → r3 = GMT2 / (4π2)

r3 = (6.674 × 10-11 × 6.0 × 1024 × 54302) / (4π2) ≈ 2.99 × 1020 m3 → r ≈ 6.687 × 106 m

Step 2 (Altitude): h = r - R = 6.687 × 106 - 6.37 × 106 ≈ 3.17 × 105 m

Answer: 317.4 km
🕹️Explore
Test your predictions in a simulation.
Interactive
​Use this interactive to explore how gravity affect orbital motion
  1. When the site loads, it defaults to a sun-earth system.
  2. Turn on the gravity, velocity and path option, and press PLAY
  3. You will find the earth will go in more or less a circular orbit. (In reality, the path is slight eccentric ie an ellipse)
  4. Now adjust the earth mass. You will find that it has no effect on. the orbit. This is consistent with the orbital velocity formula.
  5. Alter the mass of the sun. You will now find that the path of. the planet will either become an ellipse, or leave the orbit all together. (Determine the conditions for each for this to occur)
  6. You can repeat this with the earth - satellite system
  7. Now examine the sun-earth-moon. This is a little more complex, but you can see the effect of changing masses on the orbit of the moon relative to the Sun