- #1
mindcircus
- 11
- 0
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!
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!