The orbit formula
For a circular orbit, gravity provides exactly the inward pull needed to curve the path: G x M / r^2 = v^2 / r. Solving gives v = square root of (G x M / r), where G x M for Earth is about 3.986 x 10^14 cubic meters per second squared and r is Earth's radius (6,371 km) plus the altitude.
The time for one orbit is the circumference divided by the speed: T = 2 x pi x r / v. This model assumes a perfect circle and ignores air drag, which slowly pulls down the lowest orbits.
Try it: drag from 200 km to 2,000 km. The speed falls only from 7.8 to 6.9 km per second, but each lap grows from about 88 to 127 minutes. Then go to 35,800 km and the period reaches about 24 hours.
About the picture
Distances in the scene are squeezed on a logarithmic scale so low orbits and geostationary orbit both fit on screen. Animation runs about 480 times faster than real time, and Earth turns at the same sped-up rate so you can compare.