U3AOS2 Topic 3: Orbital Motion

This section is basically the same as circular motion but with just a few extra things added on. First off we have the acceleration formula in the forms

\[ a = \frac{v^2}{r} = \frac{4\pi^2r}{T} \]

Where $ T $ is the time it takes for 1 full orbit in seconds, $ a $ is the acceleration in $ m s^{-2} $, v is the velocity in $ m / s $, and r is the radius between the objects' centres in $ m $.

Then similarly, the velocity equations are:

\[ v = \frac{2 \pi r}{T} = \sqrt{\frac{GM}{r}} \]

Where $ G = 6.67 \times 10^{-11} \frac{N m^2}{s^2} $ is the universal gravitation constant and M is the mass of the attracting object in $ kg $.

Universal Formula:

From these 2 equations, you can derive a universal formula to solve for all variables in a orbital motion problem is described below. Since everything has to be balanced, this relationship will hold true for all questions you will have around this topic:

\[ 4\pi^2 r^3 = GMT^2 \]

If you have difficulty rearranging questions, it might be best to practice or write down the following forms for each variable You can also build up to the bigger formula by working with the first 2 smaller but more restrictive equations and building up.

A key relationship you can derive from the equations is that if:

\[ r \uparrow then T \uparrow but v \downarrow \]

It's important to understand that MASS DOES NOT AFFECT RADIUS OR PERIOD!!

Zero Gravity

'zero gravity' is often called free fall. It's when there is 0 normal force. This is because the ship and the astronaut are being accelerated at the same rate.