Calculating R(z) for a Coin in a Funnel | Homework Help

[jaykay99]

Homework Statement

I want to calculate the function R(z) (R=radius, z=altitude) of a funnel in which a coin( if it has no friction and only gets a horizontal speed v) always stays on the same z-coordinate.
the radius of the coin is r and r<<R.
What function has the funnel and how would a sphere behave if it has the same mass and the same radius and gets the same speed.

You can see how they work when u watch this:
http://www.youtube.com/watch?v=Loc48W0VC6o&feature=rec-LGOUT-exp_fresh+div-1r-3-HM

Homework Equations

I think it is F_z=F_g.

pls help me

The function R(z) for a pointmass is e-function but what is it for a coin?

 

[tiny-tim]

hi jaykay99!

(is v the same for all values of z?)

you need to do F = ma for the coin (assume it's a point mass) …

then put tanθ = dz/dr, and solve

 

[jaykay99]

That is clear how to calculate with a point mass.
The solution is R(z)=R_0*e^sth.

But i want to know how to calculate with a coin.

The rotation also has an effect but i don't know what.

For every speed(only horizontal) you have an z-coordinate on which the coin stays.

 

[tiny-tim]

you mean the coin is rolling on its edge (my computer isn't fast enough to show the video )?

that won't make any difference to the speed (the F = ma equation is the same), but it will affect how the coin leans, and whether it rolls "straight" …

maybe it even gives stability to the coin?

try doing the torque equations (i haven't), and see what they say ​

 

[Mindscrape]

tiny-tim, the "funnel" is one of those wishing wells that has a varied curvature, that you drop pennies into.

How complicated do you want to get? How did you get the solution for the point-mass? What you did in that solution will be able to tell anybody coming into help what assumptions you've made.

 

[jaykay99]

I have the solution as an attachment. it is a pdf.

But how to calculate with a coin?

 

[jaykay99]

Can no one help me?

 

[jaykay99]

help me pls

Has it to do with the torque?

I need an approach.

Can i say the velocity of the masspoints on the coin is not the same and therefore the centrifugal force also is not the same at the masspoints?

help me pls

 

[Mindscrape]

I would suggest using a Lagrangian with an equation of constraint, that is if you know how to use a Lagrangian. You definitely don't want to use centrifugal force because there's really no need. In any case that you do, you will want to use the moment of inertia of a disk .5MR^2, and probably use conservation of energy to your advantage in some way (which is what a Lagrangian does).

You could, however, take a similar approach to what you did the first time, and simply factor in moments of inertia. I have a feeling that it will get messy to use torques and moments of inertia with an inertia tensor.

If none of this rings a bell with you, I don't really know what to tell you.

 

[jaykay99]

Thank you for your answer!
I found sth. in the internet: http://www.spiralwishingwells.com/guide/physics.html
The form of a funnel is -1/x^2.
But it doesn't help me.
I don't know the lagrangian but i know the other moments... but i don't have any approach / idea how to start with my calculation. Can anyone help me there pls. is it that complicated?
has anyone a equation which i could use to start?

 

[jaykay99]

can anyone help me with this task?

What function Z(r) must a funnel(radius R) have if a coin(radius=r and mass=m) could rotate in that with an only horizontal velocity and the coin always stays on the same altitude(z-coordinate)? R>>r

 

[Mindscrape]

What level of physics are you in? Is this for an assignment, or personal interest? Really the best way to do this problem would be with a Lagrangian that gives the equations of motion for all generalized coordinates (such as Z, which I am betting will depend on more than just r). Especially because there is a curve that the coin is constrained to roll on, a minimum variation principle such as a Lagrangian is the only sane way to go.

After that, there will be some nasty second order ODEs that probably have to be solved numerically. At this point, we would then want to take the second order ODEs and make them into first order. From then on, we'd have to solve the, most likely, coupled ODEs with a numerical tool like Runge Kutta.

 

[jaykay99]

Wow that sounds quite difficult. It is a personal interst.
Of cousre Z depends on r, g, and v. I know that but g and v are constant.

It would be very nice if you explain it to me with the Lagrangian.
I know a bit of the Lagrangian and hope i can follow you when u pls explain it to me, this would be very nice!

 

[Mindscrape]

Hey, jaykay, yeah, I can try to explain the Lagrangian for this problem. Give me a bit to see how it works out. I'll try to get back to you sometime between now and the end of the weekend, that is, of course, unless someone else wants to jump in.

 

[jaykay99]

Can i also perhaps use the Precession?
Is that right?

 

[Mindscrape]

Let's start somewhere else first. What inertia are you looking to use? A coin should be modeled pretty well by assuming it's a "thin disc."

The next step would be to get an equation of constraint. Have you done much vector calculus? The concepts of constraints and undetermined multipliers carry over into lagrangian mechanics.