How Is the Equation of Constraint Derived for a Disk Rolling Inside a Parabola?

[mindcircus]

A disk of radius R rolls without slipping inside the parabola y=a*x^2. Find the equation of constraint. Express the condition that allows the disk to roll so that it contacts the parabola at one and only one point, independent of position.

I know the equation of constraint:
On the disk, s=R*theta.
So ds=R*dtheta
But ds is also equal to square root of (dx^2 +dy^2)
Pulling out a dx, ds=sqrt(1+(dy/dx)^2)
I know dy/dx=2ax
So sqrt(1+4a^2x^2)dx=Rdtheta

Actually, I'm not sure what to do with this. Integrate? It gets kind of messy, and I don't think I'm doing it correctly. But once I get the simplified equation of constraint, I set this equal to the function y=a*x^2?
I think I have to use Euler's equation in here somehow...but I don't see how it's relevant.

Thanks to any help!

 

[turin]

A disk of radius R rolls without slipping inside the parabola y=a*x^2. Find the equation of constraint. Express the condition that allows the disk to roll so that it contacts the parabola at one and only one point, independent of position.

I don't think I understand the question (especially the second part). I'll take a stab at the first part though.

I'm interpretting the background as 3-D. 2 background dimensions are used to describe the center of the disk. Since "without slipping" is specified, I have a feeling that the problem is asking for this to have something to do with an explicit constraint equation. So, I get another coordinate for the disk, which I will take to be an angular orientation (of some arbitrary point to some arbitrary reference line that passes through the center).

Here's where the statement of the problem confuses me. It asks for "the equation of constraint," to me implying that there is only one. However, I see that there should be two, since there are 3 possible degrees of freedom (3 dimensions), but only one allowed by the requirement that the disks rolls along the parabola (1 constraint on the 2-D position of the center of the disk) and does so without slipping (another constraint on the angular orientation).

My math skills aren't so great, but this is how I would approach the problem if I could think right now:
The center of the disk must be a shortest distance R away from the parabola at all times. This gives (I'm guessing) another parabola "that fits inside of" the one given. This is what I can't think of how to calculate. The other equation of constraint it looks like you have (or almost have) figured out. Essentially, you need to relate the arc length along the given parabola to the angular orientation of the disk as a constraint equation. Of course, I think it is probably inappropriate to do this in terms of the given parabola. I think you should do this in terms of the position of the center of the disk, which brings me back to square one.




EDIT:
Aha! Right after I hanged up, I understood the second part of the question. It is basically asking for a parabola that doesn't "pinch" the disk. This amounts to a relationship (inequality) between R and a.

 

[M98Ranger]

The equation of constraint is a mathematical expression that represents the relationship between the variables involved in a system or problem. In this case, the disk rolling without slipping inside the parabola has a constraint that ensures it contacts the parabola at one and only one point regardless of its position.

To find the equation of constraint, we can start by considering the motion of the disk. As it rolls without slipping, the distance traveled by any point on the circumference of the disk (s) is equal to the product of its radius (R) and the angular displacement (theta). This can be expressed as s = R*theta.

Next, we can consider the parabola and its equation y = a*x^2. To ensure that the disk contacts the parabola at one and only one point, the distance traveled by any point on the circumference of the disk (s) must be equal to the distance between that point and the parabola (y). This can be expressed as s = y.

Now, we can substitute the expression for s from the first equation into the second equation to get R*theta = y. To eliminate the variable theta, we can differentiate both sides with respect to x, keeping in mind that theta is a function of x. This gives us R*dtheta/dx = dy/dx.

Next, we can use the chain rule to express dtheta/dx in terms of dy/dx. Since s = R*theta, we can rewrite it as theta = s/R. Differentiating both sides with respect to x, we get dtheta/dx = ds/dx * (1/R).

Substituting this expression for dtheta/dx into our previous equation, we get R*ds/dx * (1/R) = dy/dx. Simplifying, we get ds/dx = dy/dx.

Finally, we can express ds in terms of dx and dy using the Pythagorean theorem, which gives us ds = sqrt(dx^2 + dy^2). Substituting this into the previous equation, we get sqrt(dx^2 + dy^2)/dx = dy/dx. Rearranging, we get dx/sqrt(dx^2 + dy^2) = dy/dx.

This is the equation of constraint for the system, which represents the condition that ensures the disk contacts the parabola at one and only one point, independent of its position. We can simplify this equation further by using