Coin rolling on another coin, when will free fall occur?

[dwinsemius]

Homework Statement

Doing this from a memory of a final exam question from freshman physics that I could not solve 45+ years ago. A coin of radius R is poised at the top of another (fixed) coin of radius 2R. If it rolls with no slip, at what point will the top coin leave the lower coin. (I got an A in the course so my failure to solve this question did me no lasting damage, but that difficulty has nagged me for decades.)

Homework Equations

:[/B]

The Attempt at a Solution

: [/B]
I set up the equations of motion for the coin in terms of the angle the radius of the coin to the contact point, but I was never able to figure out what to use as the condition for "launch". I suppose I might have set up an equation for the angular velocity assuming there was no loss of contact and that could be used to define an equation for the x-component for the COM. The come up with an expression for differential of x-component of a parabolic trajectory of the COM of the coin being greater. A couple of years ago I posed a similar question on some newsgroup and all I got was a reply saying they thought that Halliday and Resnick had a worked example problem. I even bought a copy and the edition I got had no such solution. Has anyone solved a similar problem.

 

[Jonathan Scott]

Interesting question.

I think that the condition for "launch" is simply when the acceleration of the center of mass of the coin (v²/r) needed to keep it following the circular curve of the lower coin exceeds the component of the gravitational acceleration of the coin in the radial inward direction (g times the cosine of the angle from vertical).

 

[gneill]

... or to express it another way, the rolling coin enters free-fall when the normal force between it and the fixed coin becomes zero.

 

[Carson Birth]

... or to express it another way, the rolling coin enters free-fall when the normal force between it and the fixed coin becomes zero.

Can confirm, had a question basically identical to this on a homework assignment a couple weeks ago :D:D

 

[Jonathan Scott]

@dwinsemius, did that help? The main thing is to consider tangential and radial components of the forces or accelerations, not x and y.