# Appearing To Reverse Object's Rotation

Can it be done, and if so, how does one you explain mathematically the ability to cause a rotating object to appear to change the direction of rotation? I believe it has something to do with angular momentum.

The thought experiment:

I have a satellite rotating around the earth in geosynchronous orbit. If the satellite approaches the planet will the earth appear to change its direction of rotation from the perspective of the satellite? At what point in its distance or angular velocity?

Second thought experiment:

In a similar illusion to the tires of a car in a movie, at what frame rate would a camera need to be filming the earth from a static position in order to have the earth appear to have a rotation from east to west?

## 1 Answer

Assuming your rotating object (e.g. the Earth) is rotating at a steady speed the only way to change it's apparent speed of rotation is if you're rotating around it.

You give the example of a geostationary satellite. This rotates around the Earth at the same angular velocity as the Earth rotates, so the Earth appears to be stationary (hence the name "geostationary"). Any satellite orbiting farther away than the geostationary orbit has an angular velocity lower than the Earth's so it will see the Earth rotate west to east. Any satellite orbiting closer than the geostationary orbit has an angular velocity higher than the Earth's so it will see the Earth rotate east to west.

There is an unrelated effect: if you hover above the North Pole you'll see the Earth rotating anti-clockwise. If you fly over to the South Pole and hover there you'll see the Earth rotate clockwise. Does this count as reversing the rotation?

Finally (and this is a nice question :-) if you film the Earth at a frame rate of one frame per 24 hours it will appear to be stationary because between each frame the Earth rotates round to the same place it was at the last frame. If you increase the frame rate the Earth will appear to rotate east to west (i.e. in reverse) until at one frame every 12 hours each frame shows 180° rotation so you can't tell which way the Earth is rotating. If you increase the frame rate still further the Earth starts rotating west to east (i.e. the normal direction).

The reverse happens if you decrease the frame rate. The Earth will appear to rotate west to east until the frame rate is one every 36 hours at which point the rotation reverses.

These frame rates are far slower than films use, but that's because wagon wheels rotate much faster than the Earth does.

Response to comment

The angular velocity of a satellite in orbit round the Earth is calculated by matching it's acceleration to the gravitational acceleration of the Earth. Suppose the satellite is orbiting at a distance $r$ and an angular velocity $\omega$, then the acceleration towards the Earth is $r\omega^2$. The gravitational acceleration is simply given by Newton's law of gravitation $GM/r^2$ and equating these gives:

$$r\omega^2 = \frac{GM}{r^2}$$

or:

$$\omega = \sqrt{\frac{r^3}{GM}}$$

If the angular velocity of the Earth is $\omega_e$ = 2$\pi$/(seconds in a day = 86400), then the apparent angular velocity at a distance $r$ is:

$$\omega_{apparent} = \sqrt{\frac{r^3}{GM}} - \frac{2\pi}{86400}$$

If you put in the geostationary orbit distance, $4.2164 \times 10^7$m (distance from the centre of the earth not its surface), then you'll get $\omega_{apparent}$ = 0 as you'd expect. Increase $r$ and $\omega_{apparent}$ gets increasingly positive while decrease $r$ and $\omega_{apparent}$ gets increasingly negative. So $\omega_{apparent}$ does indeed change sign at the geostationary orbit.

• Thanks for the thorough answer. First off, the pole question is a nice addition but I was looking for the east to west switch. So the first thought experiment was correct in assuming that a satellite in geosynchronous orbit(22,236mi) then moving closer to the earth, increasing its angular velocity, would, from the satellite's perspective cause the earth to rotate backward, east to west? Is there an equation, or series of equations, that can show that? Does the earth's angular velocity switch sign when the satellite crosses the threshold of the geosynchronous orbit? – Peregrine Dec 5 '12 at 9:30