Speed in Circular Motion (v = 2πr/T)

This is nothing more than speed equals distance over time, applied to a circle. One complete trip round covers a circumference of $2\pi r$, and it takes one period $T$, so the speed is $v = 2\pi r/T$. Everything difficult about circular motion lives in the direction of the velocity, which is changing constantly; the magnitude is this piece of grade-school arithmetic and nothing more.

The International Space Station orbits about 410 km above the surface, and Earth's mean radius is 6371 km, so its orbital radius is $r \approx 6781$ km. It completes one orbit in roughly 93 minutes, which is 5580 s. That gives $v = 2\pi \times 6.781\times10^6 / 5580 \approx 7630$ m/s, or 7.6 km/s — about 27 500 km/h, and the reason its crew see sixteen sunrises a day.

Combine it with the two neighbouring pages and a lot falls out. Since $\omega = 2\pi/T$, this equation is exactly $v = \omega r$ with the period substituted in. Put it into the centripetal acceleration $a_c = v^2/r$ and you get $a_c = 4\pi^2 r/T^2$; set that equal to the gravitational acceleration $GM/r^2$ and rearrange, and $T^2 \propto r^3$ drops out — Kepler's third law, derived in three lines from a circumference and Newton's law of gravitation.

The radius is measured from the centre of rotation, and for orbits that means from the centre of the Earth, not from the ground. Using the ISS's 410 km altitude as $r$ instead of its 6781 km orbital radius understates the speed by a factor of about four and is the most common way this calculation goes wrong. The same principle applies on a smaller scale: for a point on a flywheel, $r$ runs from the shaft axis, and for a car on a banked track it runs to the centre of the curve, not to the inside edge of the road. Two more. $T$ is the time for one full revolution — a rotation rate given in rev/min has to be inverted first, $T = 60/N$ seconds, so 1800 rpm is a period of 0.0333 s, not 1800 of anything. And this describes uniform circular motion. A real planetary orbit is an ellipse on which the speed varies continuously, fastest at perihelion and slowest at aphelion; $2\pi r/T$ with a mean radius gives an average, not the speed at any particular moment.