Angular frequency and angular velocity in UCM and SHM

[al_famky]

this is another conceptual question, so sorry for not following the template.
I thought that angular frequency and angular velocity shouldn't be the same thing, but after going through some of the forum threads, there was the idea: "the only difference between angular frequency and angular velocity is ---the former is a scalar and the latter is a vector. They have the same magnitudes. "
which got me quite confused, because in general simple harmonic motion, angular frequency is constant and angular velocity is a varible...as in a pendulum or other motions.
so i was wondering if someone could enlighten me on the matter? Thank you.

 

[BruceW]

These words can have different meanings, which is why it can be confusing at times. You are correct that the speed of a pendulum in the angular direction changes with time. And now, angular frequency, well it depends how you define it. If you define it as 2PI/T then yes, you are correct that for SHM, the angular frequency will be a constant. But if you define it as the speed in the angular direction, divided by the length of the pendulum, you can see that it is a variable.

So I think that you have the concepts correct. It is where people are using different definitions for the same word which has confused you. This does happen quite a bit in physics unfortunately. So you just need to be careful which definition is being used. And when you get really comfortable with a subject, you can often guess which definition a person is using, even though they don't explicitly state it.

 

[mukundpa]

In an ideal SHM nothing rotates or circulates. The motion of the particle is in straight line, to and fro about the equilibrium position. As the motion is periodic and the displacement, velocity and acceleration are sinusoidal functions of time, we can represent this motion as the projection of circular motion of a (imaginary) particle on a straight line. Thus every simple harmonic motion can be referred as projection of circular motion of an imaginary particle called reference particle. Angular velocity of this reference particle is called angular frequency. It has noting to do with the angular velocity of the pendulum bob.
If we consider oscillations of a spring mass system these are simple harmonic. It will have angular frequency equal to sq.rt (k/m) but there is nothing like angular velocity.

 

[al_famky]

Thank you very much, BruceW and mukundpa, it was really enlightening reading your explanations!

 

[Nickie]

Thank you for your question. It is a common misconception that angular frequency and angular velocity are the same thing. While they both involve rotation, they have different definitions and applications in the context of uniform circular motion (UCM) and simple harmonic motion (SHM).

Angular frequency, denoted by the symbol ω (omega), is a measure of the rate at which an object rotates or oscillates around a fixed point. It is defined as the number of complete rotations or cycles per unit time and is expressed in units of radians per second (rad/s). In the case of UCM, where the object moves in a circular path at a constant speed, the angular frequency remains constant throughout the motion.

On the other hand, angular velocity, denoted by the symbol ω (omega) or ω̇ (omega dot), is a vector quantity that describes the rate of change of angular displacement over time. In other words, it is the change in angular position per unit time and is expressed in units of radians per second squared (rad/s^2). In the context of SHM, where the object moves back and forth along a linear path with a restoring force, the angular velocity varies as the object reaches its maximum displacement and changes direction.

In summary, while angular frequency and angular velocity have the same magnitude in some cases, they have different definitions and implications in the context of UCM and SHM. Angular frequency remains constant in UCM, while angular velocity varies in SHM. I hope this clarifies the difference between these two concepts for you.