Pendulum Question. Details Below?

Question

For a simple pendulum, what is the distinction between the angular velocity and the angular frequency? Which is constant and which is variable?

Answer 1

I'm afraid Hannah has it exactly backwards.

"Angular frequency" is something that applies to anything that undergoes simple harmonic oscillation (like a pendulum). When something oscillates at a rate of "x" cycles per second, it's sometimes convenient to convert "cycles" to "radians" (because it makes some of the analysis easier). When the frequency is expressed in radians per second, it's called the angular frequency. When something moves in a circle, a "cycle" is equal to 2π radians, so that is the conversion factor to use, even if the motion is not circular. So if the pendulum's oscillation period is "T" seconds, then its frequency is "1/T" cycles per second, and its angular frequency is 2π times that; namely "2π/T" radians per second.

To sum up: The angular frequency is an expression of the rate of oscillation, and always equals 2π/T, where T is the period of oscillation. In the case of a pendulum, the oscillation period is constant, so the angular frequency is constant.

The angular _velocity_ is also expressed in radians per second; however (in the case of a pendulum) it refers to the rate at which the pendulum's angle is changing at any given moment. It happens to equal v/r, where "v" is the speed of the pendulum's bob; and r is the length of the pendulum. Since "v" changes (it goes faster at the bottom of the swing, and slows to a momentary stop at the top), the angular velocity is variable.

Answer 2

The angular frequency is variable since it is equal to sqrt(g/L) so it depends on the length of the pendulum. The angula velocity is constant when you don't take things like air resistance into account. Hope this helps :)