Exploring Simple Harmonic Motion in Physical Pendulums

[cjavier]

The physical pendulum is an object suspended from some point a distance d from its center of mass. If its moment of inertia about the center of mass is given by:
I= I_cm + Md²
where d is the distance from the pivot to the center of mass of the pendulum.

Consider that some odd-shaped physical pendulum of mass M is suspended from some pivot point and displaced through a given angle θ, then released. If the pendulum has a moment of Intertia I about the pivot, then the differential equation describing its subsequent motion is
I^d2θ/_dt² = -Mgdsinθ

a) Use the above info to justify that for a sufficiently small angular displacement, the physical pendulum oscillates with simple harmonic motion with angular frequency
ω = √(^mgd/_I)

SO: I know that I have to follow the argument for a simple pendulum to justify the solution for the physical pendulum. I think that is involves torque, angular acceleration, and/or moments of inertia. I am not sure how to fully justify the angular frequency equation.

 

[rude man]

What is the differential equation you know for a simple pendulum?

 

[cjavier]

What is the differential equation you know for a simple pendulum?

I'm confused if this is an actual question or one that is supposed to make me think.

The differential equation for the motion of a pendulum is I^d2θ/_dt²

 

[haruspex]

I^d2θ/_dt² = -Mgdsinθ

a) Use the above info to justify that for a sufficiently small angular displacement, the physical pendulum oscillates with simple harmonic motion with angular frequency
ω = √(^mgd/_I)

The DE above is not SHM. Your first step is to turn it into a DE for SHM by doing an approximation that's valid for small θ. Do you know a suitable approximation?

 

[rude man]

I'm confused if this is an actual question or one that is supposed to make me think.

The latter.

The differential equation for the motion of a pendulum is I^d2θ/_dt²

That is not an equation. Where's the rest of it?