Determining Equations of Constraint for Natural Motion

[Shackleford]

I assume you want to equate theta and phi somehow that would express the "natural" motion of the system. In this case, you could equate the length of the inside of the cylinder with some multiple of the circumference of the sphere as it travels along there. You could also equate the arc length of the sphere along the cylinder with the upper arc length the center of mass of the sphere traverses (R - rho).

In general, how do you determine the "best" equations of constraint?

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-11-06232419.jpg?t=1289104004

 

[Mindscrape]

Yes, the second suggestion you made sounds pretty good to me, at least if I understood you correctly. Put it into an equation and we'll see.

 

[Shackleford]

Yes, the second suggestion you made sounds pretty good to me, at least if I understood you correctly. Put it into an equation and we'll see.

f(theta, phi) = (R - pho)*theta - pho*phi = 0

If that's the best one, how do I know that?

 

[Mindscrape]

Yes, that is right. Um, well, I suppose you could do the first one. I like working with center of mass, so that sounded good to me. I guess try both out and see how they compare.

 

[Shackleford]

Yes, that is right. Um, well, I suppose you could do the first one. I like working with center of mass, so that sounded good to me. I guess try both out and see how they compare.

The first case would be R*theta = n*phi*rho. The equations are of course different. The second one is better. I guess that's because concerns the motion of the sphere, the thing that's actually moving, not just relating to some other object's quantity.

On a side note, I'm having a little bit of difficulty in their manipulation of the partials and total derivatives.

For 7.61, are partial x-sub-i and total x-sub-i equivalent since the only other variable in f is t? Also, are partial t and total d in the denominators equivalent? If so, of course, A = -A implies A = 0.

http://s111.photobucket.com/albums/n149/camarolt4z28/?action=view&current=1-1.jpg

Same thing for 7.100. For 7.101 on the left, did the total time derivatives in the numerator and denominator "cancel"?

http://i111.photobucket.com/albums/n149/camarolt4z28/2-1.jpg?t=1289109838

Same for 7.125.

http://i111.photobucket.com/albums/n149/camarolt4z28/3.jpg?t=1289109976

I understand if there's more than one variable the partials and totals are not the same.

http://i111.photobucket.com/albums/n149/camarolt4z28/4.jpg?t=1289110007

 

[Mindscrape]

Hmm, it's hard for me to tell what's going on in the context of your photos. Are you by any chance using Thornton and Marion (kind of looks like it)? I've got that book.

For the derivative with respect to time you have to worry about chain rules of variables depending on time. With derivatives with respect to coordinates it just like any common derivative.

 

[Shackleford]

Hmm, it's hard for me to tell what's going on in the context of your photos. Are you by any chance using Thornton and Marion (kind of looks like it)? I've got that book.

For the derivative with respect to time you have to worry about chain rules of variables depending on time. With derivatives with respect to coordinates it just like any common derivative.

Yes. It's Thornton and Marion chapter 7. It doesn't appear to be the best classical mechanics book, but maybe that's just me.

Do you understand my questions? The larger context really isn't that important. It's their mathematical operations that are fuzzy to me. I hope that makes sense.

 

[Mindscrape]

Ah, no, partial and full derivatives are not equivalent. 7.61 part two is just saying that the total derivative of df/dt is what's written in 7.61 part one. In general, I like to use tree diagrams to help my differentiation.

In 7.101 the derivatives just cancel. Physics is pretty bad about it's formalism for derivatives. I couldn't tell you why those cancel out, it's a math question I don't know.

7.125 if you clear up the first one, this one should get cleared up.

Honestly, I wouldn't care too much about where this stuff comes from, just that you know how to use it. That is, of course, unless you want to be a theorist, blech...