# How can the contact point of rolling body have zero velocity?

They say that for a rolling body, the velocity of the contact point is zero. I'm not getting this. How can it be zero when it's in continuous motion?

• The statement is only true if the wheel rolls without slipping. If the wheel and surface are frictionless then the wheel can spin away in one spot without moving. In fact the velocity of the wheel where it touches the surface will be negative. – Paul Linsay Apr 6 '15 at 15:52
• The velocity of the contact point is zero. Here the contact point is considered a point on the surface not a point of the rolling body. – Paracosmiste Apr 6 '15 at 17:16
• Think about a point on the top of the wheel - it has 2x the velocity of the entire vehicle. It's opposite to the bottom of the (rigid) wheel. Using trigonometry, this means the bottom must have the same speed differential, but in the opposite direction: top 2x = 1x + 1x, bottom 0x = 1x - 1x. – Jacob Krall Apr 6 '15 at 22:59
• If the velocity of the contact point would be non-zero, you would have slip (the contact point is moving relative to the ground). – Sanchises Apr 7 '15 at 11:11
• Further, consider a point on the edge of a train wheel. In the most extreme case, when it's below the top of the rail, it's moving backwards. – Steve Jessop Apr 7 '15 at 16:36

What luck! Just yesterday I was thinking about this exact same phenomenon whilst watching the film 'The Imitation Game'; the title sequence contained a moving tank.

When I was little, I used to observe this all the time; not in wheels however, but in caterpillar tracks:

Tank caterpillar tracks http://s3-ec.buzzfed.com/static/2014-04/enhanced/webdr06/21/15/anigif_enhanced-14006-1398107624-1.gif

Notice how, when a segment of the track touches the ground, it just stays there, in exactly the same spot? Obviously, its velocity must therefore equal 0, as it contacts the ground.

It was not until more recently though that I extrapolated this feature of caterpillar tracks to wheels; a wheel is just a squished together caterpillar track, if you start with a caterpillar track, and continue reducing its length, you'll eventually have a wheel.

Because any point on a caterpillar track of any size is stationary when it contacts the ground, the single point on a wheel must also be stationary as it contacts the ground.

So, the wheel is constantly moving, but the points on it accelerate, decelerate, stop, start, at different times and at different rates.

• I feel that a better animation of a tank tread could be found (the focus on this one is crushing the ... whatever that is that's being crushed, and the best angle of the treads has them in deep shadow), but as far as illustrating the point, this is excellent. – KRyan Apr 6 '15 at 19:39
• @KRyan If you want to donate one; feel free! I couldn't fins any better online, which is a shame because this one (as you said) isn't very good. – theonlygusti Apr 6 '15 at 20:02
• @theonlygusti I actually did just look, and honestly I couldn’t even find the one you did, nor any nearly as good. I do hope something better can be found. – KRyan Apr 6 '15 at 20:43
• @KRyan I actually, before reading, thought that this is the main idea. Before getting crushed, this object exemplifies the opposite situation. The contact point gets non-zero velocity (for a round object), and everyone can see what the effect is. I mean it is getting pushed, but the friction is too great. Getting rolled had much smaller movement resistances. – luk32 Apr 7 '15 at 12:34
• @KRyan I find it hard to believe that a video of a tank crushing anything is ever inappropriate. Ever. – corsiKa Apr 7 '15 at 21:07

The wheel is moving all the time, but each point on its circumference accelerates and decelerates all the time and when it is in contact with the ground it stops.

You can have a clear and convincing explanation watching this animation showing how each point on the wheel describes a cycloid

update

But remember that the ever-changing acceleration of each point is just an illusion created in the frame of reference of the road, which is at rest. This is due to the fact that the value $k$ of the translational forward-velocity of the wheel $k$ coincides with the circumference of the wheel $k = 2\pi r\rightarrow v_w = 2\pi r$ m/s:

If you can, just imagine the car moving at the same speed and the wheel spinning at the same angular velocity but not touching the ground. Or imagine the wheel of a landing plane: as soon as it touches the ground the wheel synchronizes itself to $v= 2\pi r$ m/s.

--

An illusion of a completely different kind can be experienced by stroboscopic (or wagon-wheel) effect**

• +1 This is the correct answer. The OP's confusion about "continuous motion" and "zero velocity" being contradictory is due to these concepts applying separately in different reference frames. The remedy is to show the trajectory a continuously moving point takes in the other reference frame. – imallett Apr 8 '15 at 6:08
• Ok you do need to show slope is tending to infinity at the point where it meets ground.. – samjoe May 14 '18 at 9:48

Consider a point $P$ on the surface of the wheel. If you look at the horizontal velocity of that point in the frame of reference of the wheel (axis stationary), then for a wheel of radius $r$ with angular velocity $\omega$ that point will have horizontal component of velocity

$$v_h = r\omega\cos(\omega t)$$

The linear velocity of the wheel $v = \omega r$. If we add those two velocities together, we find that the horizontal velocity in the frame of reference of the road is

$$v = r\omega\left(1 - cos(\omega t)\right)$$

That equation shows that the velocity will be exactly zero at the point where the wheel touches the road. The graph of that equation looks like this:

The position of the wheel over time (X,Y coordinates) will end up looking like this:

Both pictures should convince you that the point on the wheel really does stop momentarily.

you wrote: "How can [the velocity of the contact point] be zero when it's in continuous motion?".

However, you should keep in mind that motion is relative and therefore your question should be actually read as: "How can the velocity of the contact point be zero relative to the contact surface, when it's in continuous motion relative to its axis of rotation?" (for example, the axis of a rolling wheel).

But that is the nature of the principle of relativity; an object can be moving relative to one object and stand still relative to another.

imagine you are driving your car on the highway; your friend in the passenger seat is moving at the same velocity as you; from your point of view your friend is not moving at all; his velocity relative to you is zero, but relative to the road he "in continuous motion"; how can that be?

To me, this is most easily seen with the teeth of a gear that is running a conveyor belt.

This is basically the optimal non-slipping wheel; it rotates, and moves along the conveyor belt (really the belt moves along the wheel, but you can imagine the opposite). Each time a tooth enters the conveyor belt's track, the wheel effectively pivots on that tooth. Clearly, for a non-negligible period of time, the point at the end of tooth has no horizontal velocity, while the wheel is pivoting about it.

Have less than 50 rep, so can't write these in comments:

The question is phrased "How can it be zero when it's in continuous motion?"

@terry's answer is essentially: From the point of view of the surface (that the body rolls on), the motion of given points on the body's circumference is not continuous. Their velocity lowers until the time of contact, at which time the velocity is zero.

Adding to terry's answer: The velocity is zero for only an instant, a "point in time", immediately after which the velocity becomes non-zero again. "Intuitively" speaking, you could say that the point never stopped because it never "took the time to be in one place for some length of time". Mathematically it did stop, had zero velocity, for a moment.

Note that depending on the shape of the rolling body, there might be more than just a single point that stops. For example, for a rolling square, the whole lower side stops.

@Volker Siegel: From the perspective of the rail, the wheel is turning arount the contact point, which is at the top of the rail. Train wheels have a flange, whose edge protrudes a few centimetres below the top of the rail. When the wheel turns, for a brief time the lowest point on the flange moves backwards.

When a body rolls on a horizontal surface without slipping the horizontal velocity of the contact point is zero. This is true not only for horizontal surfaces but any surface where the body rolls without slipping.

When does a block placed on a surface has a velocity with respect to the surface?
When it changes its position on the surface. If I push the block it will slide and change its position and so the block will move with respect to the surface.

Same applies for the contact point in rolling without slipping. If the contact point slips, it has velocity with respect to the surface. Hence, in rolling without slipping, the contact point touches the surface but never slips on it and hence no motion with respect to the surface which means the point of contact is always at rest with respect to the surface at the instant it touches the surface. See this.

If not, it would skid.

[these characters are written to reach the minimum length of an answer]

• I wish I had enough rep to downvote this... – theonlygusti Apr 6 '15 at 17:35
• On the contrary, I think this is the key insight that makes the answer intuitively obvious. – peterG Apr 7 '15 at 0:37
• There is a reason for the minimum length rule, you are supposed to give helpful answers and explain the answer. – vcapra1 Apr 7 '15 at 0:47
• Okay, sorry @theonlygusti, I didn't mean to be rude. I actually thought that this fact captures the essential. There was already a good answer from Terry with a heuristic animation, but I remember when I thought about the same problem once, I first made sense to me when I realized that if v ≠ 0 in the bottom, then the wheel would skid. Maybe I should've posted it as a comment. – pela Apr 7 '15 at 9:15
• (the above comment is also for @VinnieCaprarola) – pela Apr 7 '15 at 9:15

An intuitive way to think about this is to imagine the wheel falling with the point of contact as the pivot. To exaggerate this intuition imagine a "wheel", which is an equilateral triangle rolling by falling about a vertex before rising about the next vertex and falling again. As it is rising and falling about a vertex, the vertex is stationary with respect to the surface.

Of course, in the case of the a circular wheel, the wheel only "falls" with respect to the point of contact only instantaneously to be replaced as the pivot by the "next" point on the wheel.

Hope this helps.

A planar body moving along a single axis has two degrees of freedom. Translation of the center and rotation about the center. The combination makes each part of the body to have different velocities according to the rule $$\vec{v} = \vec{v}_{cm} + \vec{\omega} \times \vec{r}$$

Rolling by definition is motion where the velocity at the contact point is zero (or no slip condition). When taken by component the above relationship becomes the scalar equation $v = v_{cm} + r \omega$. The rolling condition is $v=0$ so only movement where $\omega =-\frac{v_{cm}}{r}$ will produce pure rolling.

They say that for a rolling body, the velocity of the contact point is zero.

Indeed. In other words, as indicated in several other anwers already:
Any particular point (or smallish piece of surface) of the wheel which, at some particular instant, makes contact with the pavement has vanishing instantaneous speed with respect to the pavement, at that instant. (Otherwise we'd say that the wheel had "slipped longitudinally" along the pavement; and this holds regardless of whether the wheel axle did move at constant speed, or not.)

How can it be zero when it's in continuous motion?

Well, it bears pointing out what is "in motion" (or at least: of what non-zero speed with respect to the pavement is found), namely: the transient "spot where rubber meets the road"
which on plane ("straight") pavement has the same instantaneous speed wrt. the pavement as has the axle. (Notably, if the pavement is not plane, then the speed of this "spot" may even be arbitrarily large in comparison to $c$, as illustrated for instance by the cycloidal drive. Accordingly one should be careful with attributing "motion" to this "spot"; and its non-zero speed may best be characterized as a kind of phase speed.)

So, an important point to appreciate is that this "transient spot where rubber meets the road" is usually not referred to as "the contact point".

You're looking for velocity of points on the wheel's circumference with relation to the ground. Imagine that at each instant, each point on the circumference is connected to the point of contact by a "lever" which is a chord of the circle that forms the wheel. At point of contact the "lever" has zero length - in other words it is the only point on the circumference of the wheel that is not on the end of a "lever".

If you've ever used a throwing stick to hurl a ball for your dog to chase, you know that the length of the throwing stick increases the velocity of the ball significantly over what you can manage with your arm alone. Likewise, the longer the chord that attaches a point on the wheel to the ground, the greater that point's velocity. The longest "lever" is the chord which forms a diameter of the circle. Therefore, the point on top of the wheel has the greatest velocity. The point of contact is connected to no "lever" at all. Therefore, it is NOT being "thrown", and its velocity must be zero!

## protected by Qmechanic♦Apr 7 '15 at 17:33

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