Harmonic motion.

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 oscilation 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

[itex]\vartheta[/itex](t)=Acos([itex]\omega[/itex]t+[itex]\phi[/itex])?

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)

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Simon Bridge Recognitions: Homework Help	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 [itex]\vartheta[/itex](t)=Acos([itex]\omega[/itex]t+[itex]\phi[/itex])?

Simon Bridge Recognitions: Homework Help	Harmonic motion. 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 Recognitions: Gold Member Science Advisor	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)