Physics R' Us

[Deathslayer7]

Ok. So i'm reading about the theory of rotational motion about a rod. This is when you take the summations of the length to be infinite and thus are doing integration.

They use an example of a rod of length L and the axis of rotation is at the center of its mass.

They have this

I= integral r^2*dm

now they have dV=A*dx

and from there, they get

dm= row(density)*dV=density*A*dx

Can someone explain. I have a understanding of what is being done, but not exactly sure.

[Deathslayer7]

What I mean to ask is how do they get dm, and why do they use dm.

[Griever]

Your thread hid rather well from me, so I'm late at giving it a crack, but here we go.


I= integral r^2*dm

now they have dV=A*dx

and from there, they get

dm= row(density)*dV=density*A*dx

Can someone explain. I have a understanding of what is being done, but not exactly sure.

I'm going to make one important analogy right off the bat that took me awhile to figure out on my own in Physics 1:

Mass = Translational Inertia (Where inertia is a measure of an objects resistance towards acceleration)

So, it only makes sense that an object's rotational inertia (I) would have to do with mass.

So, we take

http://upload.wikimedia.org/math/4/0/3/403ab94a798c5c1c64af3e569e17b603.png

which is figured out through an interesting method that I can't quite explain myself, tbh. So... we'll skip that and assume this is true.

dV is a differential unit of Volume, so, since Volume is equal to length (x) times width (y) times height (z), it only makes sense that dV = Adx, where A is the area of one side (yz) so that you may integrate with respect to a single variable and not have to deal with triple integrals (consider yourself lucky)

Mass can then be defined as mass = volume * density, so dm = rho * dV, you then replace dV with Adx and you end up with dm = rho * A * dx, which is much easier to integrate!

That's the best explanation I can give until I take Differential Equations 1 and 2, I'll let you know any better answers I find when I'm finished! :smallbiggrin:

P.S. "rho" not "row" :smallwink:

[Deathslayer7]

wasn't sure how it was spelled. And i managed to figure out how to do it for a rod on my own. :smalltongue:

Although I'm heading in tomorrow and asking the teacher to explain it again for circles and a rectangle which is cut in half along a diameter. :smallsmile:

And triple integrals. pshhh. I'm already doing double. :smallbiggrin:

[TigerHunter]

I think I dislocated my brain reading that.