Where do the spokes appear unblurred in a photograph of a rolling wheel?

[boardbox]

Homework Statement

A wheel with spokes rolls without slipping on the ground. You take a picture with a stationary camera from the side of the wheel. Because the wheel is moving and the camera has a nonzero exposure time, the spokes are usually blurred. At what points in the picture do the spokes not appear blurred?

Homework Equations

Not really equations but a couple of hints I was given.

A common incorrect answer is that there is only one point.

Hint: The key idea is the following. All spokes are moving because a) the wheel is rotating and b) the wheel CM is translating (moving parallel to the floor). When you snap a picture, during a small interval dt (the exposure time) the spokes will have moved. However, there will be some points in the picture where the spokes will not appear blurred because that particular point in space will lie on the spoke throughout the motion. Note that this does not mean that the point will lie on the same point on the spoke, the spoke is still moving, but it moves in such a way that if you look at the position of the spoke at the beginning of the exposure and at the end of the exposure, one particular point will lie on the spoke in both cases. You have to find the locus of all such points.

The Attempt at a Solution

This is what I'm thinking. As the spoke moves through space, it sweeps out an area. The points I'm interested in are the points within that area that are on the spoke at t = 0 and t = dt. Sound good?

 

[tiny-tim]

Hi boardbox!

… there will be some points in the picture where the spokes will not appear blurred because that particular point in space will lie on the spoke throughout the motion.
…
This is what I'm thinking. As the spoke moves through space, it sweeps out an area. …

No, forget areas, and just follow the hint …

at which points is the instantaneous velocity towards (or away from) the centre of the wheel?

 

[Stonebridge]

You may find this page interesting. It has a very good couple of animations combined with a bit of maths.
I'm not sure what this "common incorrect view" actually is.
The dynamics of a non-slipping wheel are simple. Only the point in contact with the road is at rest. All other points are momentarily in rotation about an axis with that point at the centre. There is no other point at rest for a firm wheel (tyres/tires that compress will complicate matters), unless you look at something like the flange on a train wheel. This is covered on the link page at the bottom.
http://www.animations.physics.unsw.edu.au/jw/rolling.htm

 

[boardbox]

Rolling without slipping implies that the bottom of the wheel doesn't translate for a moment but I'm uncertain that's what I'm looking for, mostly due the wording of the question and my instructor's hint.

I'm not looking for points on the wheel that don't move. I'm looking for where spoke on the wheel would not appear blurred. According to hint my instructor gave these points in space are going to be on the spoke for the motion but this doesn't mean that they stay on the same spot on the spoke which is why I thought areas might be useful.

@tiny-tim
The velocity at at given point on the wheel should be:
Taking r with respect to the center of mass of the wheel

V = v + (w cross r) = (x, 0, 0) + (w cross r)

V = (x, 0, 0) + (0, 0, z) x (a, b, c) = (x - bz , -az , 0)

and then look for spots where that vector points back at the center of the wheel?

 

[tiny-tim]

Hi boardbox!

(have an omega: ω and a theta: θ )

The velocity at at given point on the wheel should be:
Taking r with respect to the center of mass of the wheel

V = v + (w cross r) = (x, 0, 0) + (w cross r)

V = (x, 0, 0) + (0, 0, z) x (a, b, c) = (x - bz , -az , 0)

and then look for spots where that vector points back at the center of the wheel?

Yes, but your notation is a bit weird.

Try it like this:

V = (ωR, 0, 0) + (0, 0, ω) x (rcosθ, rsinθ, 0) …

(alternatively. use the bottom of the wheel as your centre of coordinates, since it's also the centre of revolution)

 

[Stonebridge]

I think, then, what the question means is that there are some points on the spokes whose actual direction of motion, at some time, is parallel to the spoke. As far as the camera is concerned, these will not look blurred. All points on the wheel and spokes are moving relative to the ground, and camera, except the one actually in contact; on that we are agreed.
It's the interpretation of "not blurred" that is critical.

Edited to add

OS is a spoke
Points on AB are rotating about A and are moving at right angles to AB.
At P, the motion is parallel to the spoke OS.
You need to find the locus of this and all similar points.