# Mass dropped onto rotating disk

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Jazzjohn
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How would a mass fly off a rotating disk/vane system?
Picture a flat disk of radius r with a radial vane. The disk is rotating at angular velocity w. Assume the vane is straight, starts at the center and ends at the perimeter of the disk.
A very small round mass ( of m grams) is dropped onto the disk very near the center. The vane contacts it and pushes it. The mass slides along the vane and travels toward the disk rim.
Assume no friction along the vane and disk surfaces.
At the end of the vane (at the edge of the disk), the mass is spun off the disk.

1. What direction does it travel off the disk? Intuition tells me there will be a radial force component in addition to the tangential force component.

2. If the vane is not restricted to a straight line, what shape will increase the tangential direction while reducing the radial direction? Is there a theoretical optimal shape for a given mass size?

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1. What direction does it travel off the disk? Intuition tells me there will be a radial force component in addition to the tangential force component.
There are both radial and tangential velocity components relative to the fixed surface below the disk.
2. If the vane is not restricted to a straight line, what shape will increase the tangential direction while reducing the radial direction? Is there a theoretical optimal shape for a given mass size?
If the groove curves towards the edge, then the mass may come out with minimal radial velocity component.

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Summary: How would a mass fly off a rotating disk/vane system?

Intuition tells me there will be a radial force component in addition to the tangential force component.
Your intuition needs to be tempered by logic. All contacts are assumed frictionless. A frictionless surface can only a force perpendicular and away from it. The vane extends along a radius and can only exert a force tangent perpendicular to the radius. The disk is horizontal and can only exert a force "up", perpendicular to the radius.

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Conservation of energy arguments work, even when made using the rotating frame. There is a centrifugal potential associated with a uniformly rotating frame.

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Summary: How would a mass fly off a rotating disk/vane system?

1. What direction does it travel off the disk? Intuition tells me there will be a radial force component in addition to the tangential force component.
I would do this problem using the Lagrange multiplier approach. The KE in polar coordinates is: ##T = \frac{1}{2}m \dot r^2+\frac{1}{2}m r^2 \dot \theta^2##, and this problem has no potential so ##V=0##. Then the rotating vane can be represented by the constraint ##\theta=\omega t##. This gives us the constrained Lagrangian $$L=\frac{1}{2}m \dot r^2+\frac{1}{2}m r^2 \dot \theta^2+\lambda(\theta-\omega t)$$

The Euler Lagrange equations then give us $$m(r \dot \theta^2-\ddot r)=0$$ $$\lambda-mr(2\dot r \dot \theta+r \ddot \theta)=0$$ $$-t \omega + \theta=0$$ which, assuming ##r(0)=r_0## and ##\dot r(0)=0## we can solve to obtain $$r(t)=r_0 \cosh(\omega t)$$

Summary: How would a mass fly off a rotating disk/vane system?

2. If the vane is not restricted to a straight line, what shape will increase the tangential direction while reducing the radial direction? Is there a theoretical optimal shape for a given mass size?
You could do this by changing the constraint equation, but I will leave that as an exercise for the interested reader