Harmonic Motion Lecture: Deriving Equations

[tonykoh1116]

I had a lecture regarding harmonic motion.
he also derived equation related to pendulum motion with extended object and equation is following.(motion is a simple harmonic motion)
d^2θ/dt^2+(RcmMg)θ/I=0

θ(t) = θcos(Ωt)+(ω/Ω)sin(Ωt) where Ω is defined angular frequency oscillation for all types of pendulums and ω is defined angular frequency for all linear motion such as mass and spring system.

I don't get how he derived ω(initial)/Ω...
can anyone explain to me?

 

[Simon Bridge]

Welcome to PF;
Have you found the general solution to:$$\frac{d^2\theta}{dt^2}+\frac{MgR_{cm}}{I} \theta = 0$$...in a form that does not have that $\frac{\omega_{i}}{\Omega}$ in it?

But that does not look like SHM to me.
In SHM - the frequency does not change.

 

[tonykoh1116]

are you talking about

$\vartheta$(t)=Acos($\omega$t+$\phi$)?

 

[Simon Bridge]

I don't know - was I?
That would be SHM all right.

You wanted to know about: θ(t) = θ cos(Ωt)+(ω/Ω)sin(Ωt)
Looking at it properly I see that the the equation seems to be saying:$$\theta(t)=\frac{\frac{\omega}{\Omega}\sin(\Omega t)}{1-\cos(\Omega t)}$$... which is nothing like SHM right?

 

[sophiecentaur]

What was it attempting to model? That equation of motion and boundary conditions must have come from somewhere. We need to know what the ω_i term is supposed to represent. Is it an attempt to take into account the non-linearity of the restoring force in a pendulum (the frequency is amplitude dependent and, hence it is time dependent if it is decaying, for instance)