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Rolling without slipping and velocity

  1. Apr 12, 2013 #1
    I've been having trouble with this topic for a while now. I've re-read the section dozens of times now, but I'm still not exactly clear and I was hoping for clarification on a couple of things.

    In a system of particles, I can see intuitively how the velocity of a particular particle will be the sum of the velocity of the center of mass and the velocity of the particle relative to the center of mass.

    To pick an example, a tire of the moving car will have both translational and rotational motion.

    Here's a picture to illustrate what I wrote below:

    I represent the translational velocity of the entire tire by the velocity of the center of mass, so each and every particle has the same linear velocity.

    The angular velocity depends on the position of the particle.

    If there's rolling without slipping, the bottom particle must be at rest. The only way for this to happen with vector addition is for the magnitude of the angular velocity at the bottom to be equal in magnitude of the linear velocity vector but opposite in direction. Since the angular velocity is constant in magnitude but changing in direction by position, the actual velocity of each particle is the vector sum of the linear velocity of the center of mass plus the particle's respective angular velocity vector and the top particle's velocity is twice as fast as the velocity of the center of mass.

    Assuming all of the above is correct, here's where my confusion begins...

    If the bottom particle is at rest, then there can be no kinetic friction and there's only static friction.

    Assuming that the tire is perfectly circular and the surface is perfect (not bumpy) does this mean that friction does no work in the movement of the tire? If the bottom particle has zero velocity, it isn't moving and can't be doing any work.

    Does the slipperiness of a surface come into play? I'm assuming kinetic friction won't matter, but will the coefficient of static friction be relevant?

    Also, if the wheel is at rest when it comes into contact with the surface, how does the wheel even make a displacement? What stops it from just... spinning around in place? I'm just finding it very unintuitive.
    Last edited: Apr 12, 2013
  2. jcsd
  3. Apr 12, 2013 #2

    Simon Bridge

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    Yes and no - you don't do any work to move at a constant speed (in the absence of dissipation - i.e. friction in the axles). The work is done accelerating the wheel. If you concentrate only on the bit of the wheel at zero speed - then it does not have zero acceleration does it?

    The coefficient of static friction is relevant - if you try to drive the wheel faster than the road will support it will just spin in place. The wheel is not at rest when it comes into contact with the surface - only the infinitesimal part in contact with the surface is at rest, the rest is moving as your description shows.
  4. Apr 12, 2013 #3
    Thanks for the reply,

    So if the wheel is in a state where the rotational velocity is the same magnitude as the linear velocity, friction won't be acting against the wheel.

    In an ideal situation where the wheel and road is perfectly smooth, this means that there wouldn't be a tangential force acting on the wheel, so there would be no torque? The wheel would just keep on rotating forever with the same velocity, sort of how a block would slide forever if it weren't for kinetic friction?

    This part does confuse me.

    If I'm understanding you correctly, if static friction is overcome, the wheel will just spin in place? I guess it's just due to the way I'm used to thinking about friction, but I can't help but think of it as the other way around, that if the static friction were overcome, the wheel would move.

    I think I confused myself when I was writing that question. I see what you're saying.

    Rolling while slipping would be any other case where linear velocity isn't equal in magnitude to angular velocity, so the actual infinitesimal part of the wheel making contact with the surface would slide and be affected by kinetic friction, right?
  5. Apr 12, 2013 #4
    The friction forces depend on whether the wheel is attached to anything.

    In your scenario: a free moving wheel that happens to be rotating at the correct speed to satisfy the no-slip condition with the ground, and no deformation. In this case, the wheel has two forces: (1) its weight and (2) the reaction upwards from the road. Neither of these generated a torque and so neither creates a force which would act to cause the wheel to slip. Whatever the coefficient of static friction, the amount of static friction is zero because there is no force to oppose.

    I typed a huge amount on static/kinetic friction to get a 500 server error so here's the shortened version

    static friction plays a role when the wheel doesn't slip but is being driven (torque) by a car, when you accelerate the static friction at the point of contact with the road acts in the direction of the front of the car to (a) partially cancel the torque of the engine and (b) provide a force on the wheel acting forwards, which moves the car.

    And kinetic friction happens when the wheel slips, eg. brake too fast the wheel will slip, and kinetic friction will act to try to make the wheel rotate more/decrease its linear speed, until the two are equalised again at the no-slip condition (v=rw).
    Last edited: Apr 12, 2013
  6. Apr 12, 2013 #5


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    Think of a gear wheel on a toothed rack, as a simplified model. Static friction coefficient tells you how much tangential force the teeth can transmit maximally, without being sheared off. Once you go beyond that, the teeth are destroyed and you have reduced resistance from their remains, this corresponds to sliding friction. In the case of a rolling gear there are no tangential forces at the teeth, that could shear them off.

    Note that unlike the teeth in that model, the static connections between surfaces can eventually be rebuild. But some rough surfaces will be damaged by sliding, so the static friction coefficient is reduced.
  7. Apr 12, 2013 #6

    Simon Bridge

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    What MikeyW said :)

    Experimentation and observation should help your understanding at this point.
    You should get some things to roll without slipping and with slipping and watch them carefully - bearing in mind the mathematical models you have been taught.

    There are two main situations that you will want to investigate:
    1. the wheel is being driven by an engine - a locomotive wheel is probably the best for your example.
    The wheel slips when breaking too hard - the static friction is not enough to slow down so fast - or when accelerating too fast - the static friction is less than the applied torque. You've seen this, maybe even experienced it, with automobiles: you get a lot of smoke and a tire track on the road.

    2. the wheel is rolling down a ramp.
    ... probably the easiest to investigate.

    You will also want to look at the situation, in both cases, where the wheel is moving at a constant velocity.

    Have a play - then refine your play into experiments to isolate particular aspects of the motion.
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