A wheel of radius r rolls along a curve

How does this wheel roll? The velocity at the base where the curve is is facing to the right, but the theta direction seems to imply the wheel is rolling clockwise which would mean the velocity with the curve should be facing to the left. Does anyone have clarity what is going on in the picture below?

The point of the wheel instantaneously in contact with the curve is necessarily instantaneously at rest. But which point of the wheel makes contact changes over time, as does the point on the curve contacted. The velocity vector shown refers to the instantaneous motion of where contact is being made on the curve.

haruspex said:

The point of the wheel instantaneously in contact with the curve is necessarily instantaneously at rest. But which point of the wheel makes contact changes over time, as does the point on the curve contacted. The velocity vector shown refers to the instantaneous motion of where contact is being made on the curve.

So are you saying the wheel is not slipping? The problem statement seems to imply slipping because of the velocity vector in contact with the curve at the base

annamal said:

The problem statement seems to imply slipping because of the velocity vector in contact with the curve at the base

No, you are misreading the question. When it says the contact point moves at speed v0 it does not mean that a particular bit of the wheel moves at that speed.
Imagine taking a film of a rolling wheel. In each frame, mark where it contacts the road. When you run the film, how fast does your mark move?

haruspex said:

No, you are misreading the question. When it says the contact point moves at speed v0 it does not mean that a particular bit of the wheel moves at that speed.
Imagine taking a film of a rolling wheel. In each frame, mark where it contacts the road. When you run the film, how fast does your mark move?

I took a look at the solutions and it does seem like v_O'/C is moving at velocity v0 because v_O'/C = v0*vec e_t. Equations 3 and 4 below:

annamal said:

it does seem like v_O'/C is moving at velocity v0

Sure, but O' is not the same physical piece of the wheel over time. At any given time t, it is where in space the wheel contacts the curve at that instant. I.e., the mark I described in post #4. ##v_0## is how fast the mark moves.

haruspex said:

Sure, but O' is not the same physical piece of the wheel over time. At any given time t, it is where in space the wheel contacts the curve at that instant. I.e., the mark I described in post #4. ##v_0## is how fast the mark moves.

Ok, I see that. But why isn't the velocity of the base v0 = 0 then b/c it doesn't move?

annamal said:

Ok, I see that. But why isn't the velocity of the base v0 = 0 then b/c it doesn't move?

The velocity of the bit of the wheel instantaneously in contact is zero. You can think of it as the wheel rotating about that point, if it helps.
A wheel radius r moving at speed v flat road rotates at rate ##\omega=v/r##. In time dt it moves along vdt and rotates through angle ##\omega dt##. The bit of wheel at the top has moved approximately ##r\sin(\omega dt)\approx r\omega dt=vdt## further than the middle, so is moving at speed 2v. Likewise, the bit in contact with ground has moved approximately ##r\sin(\omega dt)\approx r\omega dt=vdt## less far than the middle, so is moving at speed 0.