Is angular SHM also related to UCM ?

[ximath]

Hi all,

I have learned that linear SHM is just projection of UCM onto one axis. Is angular SHM also related to UCM ?

Moreover, I have also learned that theta = theta0 * sin(wt+P) where theta is angular displacement, theta0 is angular amplitude and P is phase constant. Where does this equation come from, or how can I derive it ?

 

[ximath]

Well, I somehow get the idea that they are related..

Thus instead, "how they are related" would be a better question actually.

 

[ximath]

I guess I need to be more precise. Assume we have a simple pendulum. We say that w = sqrt(g/L). Moreover, we say T = 2PI / w

It is quite a lot confusing for me to say period is equal to 2PI / w for a simple pendulum. First of all, if the angular displacement was 2PI and w was constant; then it would make sense to say T = 2PI / w . However, w is not constant and the angular displacement is not 2PI..

Well, if motion is analog to a uniform circular motion in which w is constant and always equal to sqrt(g/L); then it seems to be fine. But I'm not sure about that, too.

I seriously need some help here.

 

[Doc Al]

Don't confuse the angular frequency of the pendulum's simple harmonic motion (ω), with the angular speed of the pendulum itself.

 

[ximath]

Hmm, so as far as I understand, angular freq. is constant and angular speed (of pendulum) is variable.

saying w = sqrt(g/L) ; w is here angular freq. I guess.

Can we say that angular frequency is actually angular velocity of a UCM(not the pendulum) analog of our SHM in this case ?

 

[Doc Al]

Can we say that angular frequency is actually angular velocity of a UCM(not the pendulum) analog of our SHM in this case ?

I don't see why not.

x = x_0\sin \omega t

is analogous to

\theta = \theta_0\sin \omega t

(You'd have to have the radius of the circular motion represent θ₀.)

 

[ximath]

Thanks a lot!

Now, I would like to prove that

w = \sqrt{\frac{g}{L}}

I have tried:

Ftan = - mg \sin\theta

(for simple pendulum)

and

a = g \sin\theta

V = \int g\sin\theta dt 0 to t

and

X = \int V dt 0 to T/2.

Moreover;

X = \theta L

Then; using w = \frac {2PI} {T}

I would obtain a formula for w -- which hopefully would be equal to

w = \sqrt{\frac{g}{L}}

So I guess now the problem is to calculate

V = \int g \sin\theta dt

Well, you say that

\theta = \theta_0\sin \omega t

but I don't really know why this equation is correct.

 

[Doc Al]

Now, I would like to prove that

w = \sqrt{\frac{k}{m}}

That's for a mass on a spring, not a pendulum.

I have tried:

Ftan = - mg \sin\theta

(for simple pendulum)

Why the tan?

Just: F = -mg sinθ

Then use a small angle approximation and Newton's 2nd law to set up (then solve) the differential equation.

 

[ximath]

Thanks!