Quote by sophiecentaur
The (differential) equation of motion for a particle of mass m on a spring with spring constant k and a displacement x from the equilibrium position is
m d_{2}x/dt^{2} = kx
When you solve this, you find a solution of the form
x= A sin(ωt  ∅)
∅ is an arbitrary value for the phase of the oscillation.
That's where the variable ω comes from. Trig functions involve angles so ω has the dimension of an angle divided by time. Hence it's referred to as an angular frequency. There is one other point and that is that ω is in radians (as all good angles are) so is 2∏f, where f is the frequency in cycles per second (Hz).

That doesn't really help me intuitively to see where the angular speed comes from.
Also, since you solved the differential equation of motion for a particle with a spring, does that mean that the solution only applies to simple harmonic motion with springs? Is there simple harmonic motion without a spring?