Marion and Thornton Dynamics Problems Problem 7-3

[MARX]

Homework Statement

A sphere of radius ρ constrained to roll without slipping on the lower half of the inner surface of a hollow cylinder of inside radius R. Determine the Lagrangian function, the equation of constraint

Homework Equations

v= ω × r

The Attempt at a Solution

NOT HOMEWORK-SELF LEARNING

Can someone look and tell me if I am deriving the equation of constraint correctly in this problem
rest I understand

 

[BvU]

Ah, an odd-numbered exercise -- not in the solution book .

For a 1D treatment I think you're doing fine (*). You want to work towards which generalized coordinate ?

PS I'd avoid choosing the y-axis downwards...

(*) the exercise mentions a cylinder constraint, but the coordinate into the paper would make life a lot harder, so I'd stick to 1D indeed. @haruspex: do you agree ?

 

[MARX]

Ah, an odd-numbered exercise -- not in the solution book .

For a 1D treatment I think you're doing fine (*). You want to work towards which generalized coordinate ?

PS I'd avoid choosing the y-axis downwards...

(*) the exercise mentions a cylinder constraint, but the coordinate into the paper would make life a lot harder, so I'd stick to 1D indeed. @haruspex: do you agree ?

The solutions are not detailed at all and either way I don't look at them till I'm done and most of the time I take different approaches :)
The generalized coordinates are θ and Φ I can relate them and make the problem one G coordinate however since the question is asking for the equation of constraint I would have to keep it at 2 GC
I already found L correctly (not on picture)
Also We know that the sphere cannot roll into and out of the page so that immediately knocks off Z as the normal force and weight are both in the x-y plane so angular momentum L_{z is conserved-no side motion remember just like the particle on the hemisphere (in the frame of the sphere I mean)!}

 

[BvU]

The solutions are not detailed at all and either way I don't look at them till I'm done and most of the time I take different approaches :)

Don't fuzz; I was just teasing. It's hard enough to study on your own (hats off!) -- good thing PF exists .
(I do see now that the odd-numbered ones are NOT skipped in the solutions manual, only in the book itself) -- and yes, it's treated as a 2D problem + 1 constraint, Usually this is done to allow extracting the forces of constraint as in example 7.9 -- so here that allows finding $\omega$ .

I already found L correctly (not on picture)
Also We know that the sphere cannot roll into and out of the page

Agree. But the problem statement doesn't say that.

 

[MARX]

Don't fuzz; I was just teasing. It's hard enough to study on your own (hats off!) -- good thing PF exists .
(I do see now that the odd-numbered ones are NOT skipped in the solutions manual, only in the book itself) -- and yes, it's treated as a 2D problem + 1 constraint, Usually this is done to allow extracting the forces of constraint as in example 7.9 -- so here that allows finding $\omega$ .

Agree. But the problem statement doesn't say that.

not fuzzing at all i know you were kidding thanks

Don't fuzz; I was just teasing. It's hard enough to study on your own (hats off!) -- good thing PF exists .
(I do see now that the odd-numbered ones are NOT skipped in the solutions manual, only in the book itself) -- and yes, it's treated as a 2D problem + 1 constraint, Usually this is done to allow extracting the forces of constraint as in example 7.9 -- so here that allows finding $\omega$ .

Agree. But the problem statement doesn't say that.

I was not fuzzing I know you were kidding I really appreciate all your help thanks
I enjoy learning it this way to be honest with you I don't think most physics professors understand lots of these problems with all due respect-qouting my personal experience- anyways especially with physics I firmly believe 80% has to be self learned. + PF definitely helps draw back is time but definitely woth it :)

 

[MARX]

Don't fuzz; I was just teasing. It's hard enough to study on your own (hats off!) -- good thing PF exists .
(I do see now that the odd-numbered ones are NOT skipped in the solutions manual, only in the book itself) -- and yes, it's treated as a 2D problem + 1 constraint, Usually this is done to allow extracting the forces of constraint as in example 7.9 -- so here that allows finding $\omega$ .

Agree. But the problem statement doesn't say that.

and yes I meant 2GC+1 C to find force of constraint not just equation of C the question asks for that

 

[TSny]

Can someone look and tell me if I am deriving the equation of constraint correctly in this problem
rest I understand

It looks good to me. I like your way of arriving at the constraint by using relative velocity concepts.

However, it is good to check the signs in your final constraint equation. Are you choosing the sign conventions for $\theta$ and $\phi$ so that they both increase in the counterclockwise direction? By looking at your figure, if $\theta$ increases, would $\phi$ also increase or would it decrease? Is this in agreement with your constraint equation?

 

[MARX]

It looks good to me. I like your way of arriving at the constraint by using relative velocity concepts.

However, it is good to check the signs in your final constraint equation. Are you choosing the sign conventions for $\theta$ and $\phi$ so that they both increase in the counterclockwise direction? By looking at your figure, if $\theta$ increases, would $\phi$ also increase or would it decrease? Is this in agreement with your constraint equation?

Ahah
great point thanks
For that I look at the time derivatives of the angles so

• as the sphere rolls downwards the angular speed is positive (counter clockwise) hence Φ is increasing because we are approaching bottom ie max kinetic energy and zero potential
• while θ igets smaller as sphere rolls down so rate of θ is negative clockwise so yes when former is positive the latter is negative so signs match

Am I interpreting this correctly?

 

[TSny]

Ahah
great point thanks
For that I look at the time derivatives of the angles so

• as the sphere rolls downwards the angular speed is positive (counter clockwise) hence Φ is increasing because we are approaching bottom ie max kinetic energy and zero potential
• while θ igets smaller as sphere rolls down so rate of θ is negative clockwise so yes when former is positive the latter is negative so signs match

Am I interpreting this correctly?

Yes. If you define the positive direction to be counterclockwise for both $\theta$ and $\phi$, then when one of them increases the other decreases. Does your constraint equation agree with this?

 

[MARX]

Yes. If you define the positive direction to be counterclockwise for both $\theta$ and $\phi$, then when one of them increases the other decreases. Does your constraint equation agree with this?

Yes that seems to agree with my constraint equation just to save time if you think otherwise please say it this is not homework I am self learning
thanks for your help

 

[TSny]

If you rearrange your constraint equation in the form $\rho \phi = (R-\rho)\theta$, you can see that this implies that if $\theta$ increases then $\phi$ also increases.

 

[MARX]

great thank you very much understood!