Rigid Body Dynamics and the Center of Mass of a Rigid Body

Rigid body mechanics…

Rigid body? A strong object? Right?lol

Yeah. That’s right lolol

But how strong is it — it’s hella strong

So why do we study particle systems first and then move on to the mechanics of rigid bodies?

The rigid body here is an ideal object, so it doesn’t exist on Earth!!

Like, is there some freaking insanely strong megaton steel ball that never bends or deforms no matter what?

Nope, everything deforms at least a ti~~~ny bit, right?

So then, why do we learn particle systems and then learn rigid bodies?????

Rigid body: it’s said to be a particle system where relative positions are fixed, or where the distance between any two particles is always constant.

The particle systems are moving, and the fact that the distance between those particle systems is always the same means

It won’t allow even a 0.0000000000000001 deformation

OKBR

So let’s find the center of mass of a rigid body.

Previously it was a discrete configuration of particles,

but now it’s composed of continuous particles,

so we throw out the sigma symbol, and meaning-wise it’s still a sum, it’s a sum with the same meaning, but

if we use the integral, which means the sum of ‘continuous’ things,

Alright then, let’s calculate the center of mass for x, y, z.

If the rigid body is a 3D object

If the rigid body is a 2D plate

So for a solid hemisphere and a solid hemisphere shell, I’ll calculate the center of mass as practice~

This right here is a hemisphere!! It’s completely filled inside

If so, it’s way too obvious that Xcm and Ycm are 0, right? right??

Right now the O point is that spot marked with an x.^^

So the key is finding Zcm,

We chop it into thi~~~n disks at height z! Did you chop it?(in your head)

After chopping, how are we going to add them up — we’re going to multiply by the z-value, which is the position where the disk is

That z-value is called the weight

We take that z-value as the weight, multiply it by the mass of that thin thin disk, add them all up!

Then divide by the total mass, and you can think of that as Zcm

The hemisphere shell is easy too hehe

This time, instead of a disk at an arbitrary height z,

you can think of it as a ring

We multiply the mass of that ring by the weight of height z, and then

add them allll up again, and then divide by the total mass (the total mass of the hemisphere shell)!!!