Finding the instantaneous axis of rotation

[Femme_physics]

Homework Statement

http://img684.imageshack.us/img684/2763/wheelturning.jpg

In the drawing is depicted a wheel with radius R (measurements in pic). The wheel rolls on a horizontal plane, across a straight path, without sliding. At a certain moment of the movement, the center of the wheel's velocity (Vo) equals 30 m/s.

Calculate

A) The absolute velocities of point A and B in that moment
B) Mark in the drawing the directions of the velocities

The Attempt at a Solution

http://img691.imageshack.us/img691/6429/wheresp.jpg

Where P is the instantaneous axis of rotation. I can't find it since if the distance to P from the center of the circle is x, an unknown, and I also don't know Vb, then I have 2 equations with 2 unknowns!

I had a similar problem but since in that other problem I actually had both velocities I could find P. How do I find P this time?

 

[lrkr82]

I venture a guess that P is just where both objects contact; thus: PO=R
That's what I know about round things rolling on flat things :)

 

[gneill]

The point of contact of the wheel with the road has an instantaneous velocity of zero -- it's a momentarily stationary point. Draw lines from that point to the center of the wheel and to points A and B. Imagine that these are radius vectors and that they all rotate with the same angular velocity...

 

[I like Serena]

P is the point of contact of the wheel with the road.

 

[Femme_physics]

Aha! So whenever a wheel is rolling on a surface, the instantaneous axis of rotation is simply the connecting point with the ground. So, with this logic, I can find the distances to P and find out Vb and Va. Let me see.

Well, I was able to easily get Va. Va is just

http://img204.imageshack.us/img204/5959/11559339.jpg


Now Vb is a bit more complicated. I'm not seeing how I have enough info to solve for it.

http://img109.imageshack.us/img109/4949/har1.jpg

http://img804.imageshack.us/img804/2449/har2.jpg

Help/hints?

 

[Femme_physics]

No wait, I'm up to something, don't tell me anything!

 

[Femme_physics]

Got it :) Just needed to add 90 and use Law of Cosines. Problem solved :) Thanks, everyone!

 

[I like Serena]

No wait, I'm up to something, don't tell me anything!

Sorry, I can't keep shut, I just have to say:

LOL!