Rotation of wheel between two non-slip surfaces

[Brian in Twisp]

Homework Statement

A certain machine can be modeled as a wheel between two translating bodies. Point P is on the upper translating body and is moving to the left at 6m/s and Point Q is on the lower translating body and is moving to the right at 3 m/s. The radius of the wheel is .3m. Find the velocity at the center of the wheel and velocity at point B. Assume that the wheel is not slipping on the translating bodies (I didn't get the translating bodies labled on the diagram, but they are the two surfaces that the wheel is between.

Homework Equations

V_c = r ω
V_B = V₀ + V_rel + ω x r

The Attempt at a Solution

Using the relative velocity between two different frames equations, I got the velocity of the center at -3m/s i, but I my answer for the velocity of B wasn't correct, and I'm not sure where I'm erring.
My interpretation is: Since the wheel is not slipping, the point of contact is moving at the same velocity as the translating body. Looking at the upper body, P is moving at -6m/s. This should be equivelent to a wheel rolling on a fixed plane at 6m/s, ω = 6/-0.3 or 20 radians/second counterclockwise.
So I'm getting a velocity of B = -3i -6j

The problem is in a unit focusing on the kinematics of motion of solid bodies between two different frames.
For full disclosure, the problem was on a quiz that I passed, but missed this question. The instructor doesn't release solutions because people can take the test at different times.

Thanks in advance.

 

[Simon Bridge]

If you take the +ve direction to be "to the right" then point P is moving at -3m/s wrt point Q.
Which part of the wheel has the same velocity as point P?

Note:

My interpretation is: Since the wheel is not slipping, the point of contact is moving at the same velocity as the translating body.

... get a round object and roll it along the floor ... observe the point of contact between the object and the floor: what speed is the object at that point? Is that the same as the speed of the object?
In your problem, the floor also happens to be moving ... so take your round object (a can is good for this) and put it between two flat objects (large books say) and move them about. Put the can on the floor, put a book on top of it, move the book ... how far does the can move, relative to the floor, compared to the book?

Watch and learn.

 

[Nathanael]

If you take the +ve direction to be "to the right" then point P is moving at -3m/s wrt point Q.

Do you mean -9m/s?

Anyway I also disagree with your first answer of -3m/s for the motion of the center. Please show your work on that part.

 

[Simon Bridge]

Sry yes. The rest stands.

 

[haruspex]

Note: ... get a round object and roll it along the floor ... observe the point of contact between the object and the floor: what speed is the object at that point? Is that the same as the speed of the object?

Brian's statement was that the point of contact (presumably meaning the point on the wheel at the point of contact) is moving at the same speed as the "translating body", i.e. (Based on the terminology in the problem description) the flat surface with which it is in contact. I see no flaw in that.

 

[Simon Bridge]

Neither do I.
I want OP to watch something roll and use the observations to understand the problem better.

 

[haruspex]

This should be equivelent to a wheel rolling on a fixed plane at 6m/s

That would be true if the centre of the wheel were not moving: upper plane moving left at 6m/s, centre of wheel static, would be the same as upper plane static, wheel rolling along underneath it as 6m/s to the right. But the centre of the wheel is moving.

 

[Brian in Twisp]

So looking at the problem assuming both the point of contact on the top and the bottom are stationary with respect to their respective surfaces, and the relative motion between P and Q is -9m/s. Assigning p and q add the respective points of contact, I can then use the relative velocities to calculate the angular velocity with V=ω x r taking r as the distance (0.6m) from q to p. Therfore ω=15.
Now knowing ω, I can go back and, using q as my instantaneous point of zero velocity, calculate the relative velocity of the center at q-4.5i or restated back in the original frame is -1.5i. By the same logic the velocity at point B is -1.5i -4.5j.
This fits with what I saw when I rolled a couple round things across the counter under a book, and saw the wheel moving half the distance, and thus half the speed of the book.
I think I got it now, thank you all. But please let me know if I said something Thai doesn't make sense.
Brian