Physics on a 2D Circle

Question

Lets suppose I have a flat, 2 Dimensional disk of a certain weight and resides on an XY coordinate plane.
Now lets say I apply a force somewhere on the circumference of the disk. This force can be at any angle and any force. How would I calculate the direction (0-360) and speed gained (ignoring friction) and the new spin speed (using angles 0-360)?
Given the current speed, direction, rotation angle, and rotation speed, how would I calculate the outcome speed, direction, and rotation speed?

Answer 1

I can think of three possible outcomes based on the orientation of the force vector:

1. "flipping" like you would expect with a coin
2. rotation about its center like a wheel
3. translation

 

Of these three types of motion, the rotation and translation take place in the plane of the disk. Only flipping would move the disk with respect to Z.

 

Because the disk is in the XY plane, it has a unit normal vector n=[0,0,1]. The component of the force vector that is directed parallel to n is the magnitude of force that drives the flipping motion (computed with a dot product F•n). Knowing that the force is applied on the circumference of the disk (out a distance R from the center), the resulting torque would be given by the cross product T=R×(F•n). (The angle between these vectors is 90 degrees so you can simply multiply R and the value for Fn.) You can then use torque and moment of inertia for a flipping disk to calculate angular velocity/acceleration/etc.

 

The in-plane translation and rotation is determined by the remaining components of the original force vector. Simply subtract off the normal component to get the XY component: f=F-(F•n)n. We can split f into a component tangent to the disk (which drives rotation) and perpendicular to the disk (driving translational motion). 

 

The perpendicular component is directed radially from the disk's center, so the magnitude of the translational force would be the dot product of f and the perpendicular unit vector r. Then you can derive the disk's motion by f•r=ma.

 

The tangential force (running out of letters, so I'll just call it g) can be found by subtracting off the perpendicular component: g=f-(f•r)r. The magnitude of this vector will exert another torque, and you can derive the resulting angular motion again with t=Iα. 

 

 

edit: Forgot to mention, the three direction vectors in this problem (normal, radial, tangential) are all mutually perpendicular, so they will combine independently of one another to form the disk's net motion. The three vectors for a basis spanning all of 3D space (cylindrical coordinates), so any 3D force vector F can be exactly decomposed into those coordinates