Can someone confirm that a rolling coin that is falling will, at any instant, only have horizontal velocity directions that are parallel (those velocities being horizontal component velocities)?

It is counterintuitive because the coin will continually turn and travel a curved path, thus we would think that there is a vertical axis rotation with velocities of different directions tangent to a circle of an instant turn center. But I only see one direction horizontal component velocities of every part of the coin at any instant.

In Fig 1, a wheel rolls and the horizontal component velocities of every part of the wheel are oriented in one direction (top view) on the horizontal surface on which the wheel rolls except for the stationary part(s) at the contact point. Velocity arrows only indicate direction and not magnitude. Every non-zero horizontal component velocity is going to the right which is parallel to the wheel plane that defines the wheel’s rolling direction. If the wheel was stationary and falling sideways, the same thing (same direction horizontal component velocities) occurs except now orthogonal to the wheel plane. enter image description here If we combine the rolling and falling motion we get what we’ll call tumbling of the wheel. The tumbling wheel also has same direction horizontal component velocities at any instant. Figs 2 and 3 show this tumbling wheel as it travels along a straight path. enter image description here enter image description here enter image description here A better illustration of true rolling and falling motion of a wheel is shown in Fig 4. The wheel rolls and falls thus the velocity direction of the wheel center of mass is always at an angle to the wheel plane. Again, all directions of the non-zero horizontal component velocities on the rolling surface plane are parallel, at any instant. enter image description here In Fig 5, a cornering vehicle wheel has horizontal component velocities of different directions on the horizontal rolling surface plane. The velocity directions are always tangent to an instant turn center circle. A vehicle wheel changes direction by a different mechanical reason compared to the rolling falling wheel. enter image description here

  • $\begingroup$ I have never heard of this concept of “aligned velocities” and the velocities are certainly not what I would call “aligned” $\endgroup$
    – Dale
    Aug 7, 2022 at 14:59
  • $\begingroup$ Acknowledged and corrected. $\endgroup$
    – Nectac
    Aug 7, 2022 at 16:07
  • $\begingroup$ What do you mean by "oriented velocities in one direction relative to the surface on which it rolls?" (I presume orintented is a typo.) The diagram seems to show that the velocities can point in any direction through 360 degrees. Although the diagram is clearly a little misleading in that it shows all the velocity arrows as close to the same size. But it should show them as much smaller near the contact point. $\endgroup$
    – BillOnne
    Aug 7, 2022 at 16:16
  • $\begingroup$ No typos. The arrows in the figures only show direction and do not indicate magnitude. The top view arrows show velocity direction relative to the surface on which the wheel rolls. Fig 5 shows that the velocities of a cornering vehicle wheel have directions that are tangential to an instant turn center. The wheel that is rolling and falling in Figs 1-4 will always have velocity directions in the same direction relative to the rolling surface. Both wheel examples travel curved paths. $\endgroup$
    – Nectac
    Aug 7, 2022 at 20:51

1 Answer 1



A wheel that is tipped, tipping and rolling does have, at every instant in time, this feature: All of the velocities of all of the points on the wheel, are in a common plane. This is a feature of any rigid object on which there is a point, for a wheel this is the ground contact point, with zero velocity. What plane is this? It's the plane orthogonal to the instantaneous axis of rotation.

Given that there is a point with zero velocity, a rigid object always has an axis through that point which has the property that all of the velocities of points on the object are in the plane orthogonal to that axis.

If that axis is on the ground plane, say the xy plane. Then the projection of all of the velocity vectors on to the xy plane are parallel.

Is this what you are trying to say?

If, however, the angular velocity vector of the wheel has a vertical component, then the velocities (all of which are on the plane perpendicular to the axis) do not all project to a common direction on the xy plane.

Superficially, it seems like you are saying that the angular velocity of a rolling wheel never has a vertical component, which is a false statement.

It is true, however, that one can accomplish a net rotation about the z axis by only rotating about the x and y axis. But there is no reason to think that this is how a rolling wheel turns.

If you think the physics somehow makes it so that the angular velocity vector never has a vertical component, you are wrong. One can set a rolling wheel going with initial conditions corresponding to angular velocity about any axis.

  • $\begingroup$ Hey Andy, What I’m trying to say is that the horizontal velocity components of all non-zero velocities of the wheel are parallel, as you agreed with, and that the direction of those velocities is the instantaneous direction of travel of the wheel with respect to the rolling (horizontal) surface. At some next instant the direction might be different but the same description applies at that instant. So Fig 4 shows what a coin might do, which is to constantly turn, ….continued…. $\endgroup$
    – Nectac
    Oct 11, 2023 at 0:33
  • $\begingroup$ ….yet at no instant in time was there a component of rotational velocity on the horizontal plane. In other words, the wheel rolled, turned and traveled along a curved path without any direction changing torque. $\endgroup$
    – Nectac
    Oct 11, 2023 at 0:34

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