YoYo rolling on ground (checking work mostly)

[phil ess]

Homework Statement

See post #3 Assume the YoYo is a solid cylinder with mass M and radius R for all parts!

Homework Equations

Lots?

The Attempt at a Solution

Part A - Find the moment of inertia about the center of mass and the point of contact with the ground:

I_COM = 1/2 MR²
I_IPoC = 1/2 MR² + MR² = 3/2 MR² by parallel axis theorem

Part B - Calculate the torque on the Yoyo about the center of mass, and the net horizontal force.

Torque = r x F = rF since r is perp to F
F_{net x} = F cos θ

Part C - What is the angular accel. of the yoyo about its center of mass? Linear acceleration?

Torque = rF = Iα α = rF/I α = 2rF/MR²

and since a_lin = αr
a = 2r²F/MR²

Part D - Find the net torque about the instantaneous point of contact

Ok here's where I get stuck. Torque = rF sin σ , where σ is the angle between the applied force F and the r vector, right? Well no joke the geometry is complex. I can't find any way to get the angle between these two, here's what I've got so far:

σ is the angle between the two blue lines, which represent the force and the r vector. I've also extended the force line so you can see the line of action, if it helps.

Any help would be greatly appreciated!

 

[Doc Al]

Can you at least describe the problem and what you are asked to find.

 

[phil ess]

The picture you see above is the bottom half of a YoYo of mass M. The inner axle has a radius r and the outer radius is R. The string is wrapped under the axle and being pulled gently so the YoYo rolls to the left without slipping.

Im asked to find the net Torque on the YoYo about the instantaneous point of contact with the surface. Trouble is I can't find the angle between the vectors, which is what the above diagram is trying to accomplish.

 

[Doc Al]

With a bit of trig you should be able to find the coordinates of the point of application of the force, thus you can find the position vector of the force and the angle it makes with the horizontal.