Satellites and Weightlessness: Quick Revision

A brief note on Satellites and Weightlessness. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Orbital speed: gravity supplies the pull to the centre, $\dfrac{GMm}{r^2} = \dfrac{mv^2}{r}$, so $v = \sqrt{\dfrac{GM}{r}}$. The satellite's own mass cancels.
• Period: $T = 2\pi\sqrt{\dfrac{r^3}{GM}}$; higher orbits are slower and take longer.
• Energies: $K = \dfrac{GMm}{2r}$, $U = -2K$, $E = -K$. A bound satellite has negative total energy.
• Geostationary: period one day, above the equator, moving east, about $36\,000\ \text{km}$ up. Polar: low, over the poles, and the Earth turns beneath it.
• Weightlessness: in orbit, the craft and everything in it fall together, so nothing presses on anything. Gravity is still there.

The facts at a glance

A worked example

A $450\ \text{kg}$ weather satellite circles $1500\ \text{km}$ up. Use $GM = gR^2 = 4.01 \times 10^{14}\ \text{m}^3/\text{s}^2$ and $R = 6400\ \text{km}$.

1. Radius. $r = 6400\ \text{km} + 1500\ \text{km} = 7900\ \text{km}$.
2. Speed and period. $v = \sqrt{\dfrac{4.01 \times 10^{14}\ \text{m}^3/\text{s}^2}{7.9 \times 10^{6}\ \text{m}}} = 7.13\ \text{km/s}$, and $T = \dfrac{2\pi r}{v} = 116.1\ \text{min}$.
3. Total energy. $E = -\dfrac{GMm}{2r} = -1.14 \times 10^{10}\ \text{J}$.

Practice with new numbers

A program to try

The program works out the weather satellite of the worked example.

Mistakes to avoid

• The height for $r$. In every orbit formula, $r = R + h$.
• A heavier satellite needing more speed. The mass cancels; every satellite at that height moves at the same speed.
• A positive total energy. A satellite in orbit has $E = -K$, below zero.
• Raising an orbit makes it faster. It ends up slower, though it took energy to get there.
• No gravity in orbit. At $300\ \text{km}$ up, $g$ is still about $90\%$ of its value on the ground.

Check yourself

1. Does a heavier satellite need a greater speed for the same orbit?

2. A satellite's orbit radius is made $4$ times larger. What happens to its period?

3. A satellite has $3.0 \times 10^{9}\ \text{J}$ of kinetic energy. What are its potential and total energies?

4. Why must a geostationary satellite be above the equator?

5. What does a spring balance read when an astronaut stands on it in an orbiting station?