Rolling

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© 1998-2008 by Glenn Elert -- A Work in Progress
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Discussion

Rolling without slipping is a combination of translation and rotation where the point of contact is instantaneously at rest.

When an object experiences pure translational motion, all of its points move with the same velocity as the center of mass; that is in the same direction and with the same speed

  v(r) =  vcenter of mass  

The object will also move in a straight line in the absence of a net external force.


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When an object experiences pure rotational motion about its center of mass, all of its points move at right angles to the radius in a plane perpendicular to the axis of rotation with a speed proportional to the distance from the axis of rotation …

v(r) =  rω

Thus points on opposite sides of the axis move in opposite directions, points on the axis do not move at all since r = 0 there …

  vcenter of mass =  0  

and points on the outer edge move at the maximum speed …

  vouter edge =  Rω  


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When an object experiences rolling motion the point of the object in contact with the surface is instantaneously at rest …

  vpoint of contact =  0  

and is the instantaneous axis of rotation. Thus, the center of mass of the object moves with speed …

  vcenter of mass =  Rω  

and the point fathest from the point of contact moves with twice that speed

  vopposite the point of contact =  2vcm = 2Rω  


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The wheel is an extension of the foot.


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cycloids

rolling resistance

Coefficient of Rolling Resistance for Selected Interfaces
interface coefficient
bicycle tire on … wooden track 0.001
  smooth concrete 0.002
  asphalt road 0.004
  rough but paved road 0.008
Source: Analytic Cycling

 

Summary

Problems

practice

  1. Complete the worksheet on the first page of translate-rotate-worksheet.pdf. Fill each grid space with an appropriately concise answer.
       
    page 1: the questions [magnify] page 2: the answers [magnify]

    The answers are also available on the second page of the worksheet [pdf] and on this webpage [html].

  2. Release an object from rest and let it roll down an incline. Determine …
    1. the moment of inertia coefficient by timing the trip from top to bottom
    2. the critical angle past which an object will slip rather than roll down the incline

    Solutions …

    1. This problem is best solved using the conservation of energy. The rolling body starts with gravitational potential energy at the top of the ramp and ends with translational and rotational kinetic energy at the bottom. Since the object isn't slipping, it's rotational velocity is v / R.
       
      Ug =  Kt  +  Kr    
      Mgh =  1  Mv2  +  1  Iω2  =  1  Mv2  +  1  I  ⎛
      ⎝      		 v ⎞2
      ⎠
      2 2 2 2 R
      2MghR2 =  MR2v2  +  Iv2    
       
      I =  2MghR2 − MR2v2  =  ⎛
      ⎝      		 2gh  −  1 ⎞
      ⎠      		  MR2
      v2 v2
       
      Here's the coefficient. It still needs a bit of work.
       
      α =  2gh  − 1
      v2
       
      Recall that for an object accelerating uniformly from rest, its final speed is twice its average speed.
       
      v =  Δs  =  v + v0  =  v + 0
      Δt 2 2
       
      v =  2Δs
      Δt
       
      Substitute this expression into the formula for the coefficient. (We'll drop the delta symbols.)
       
      α =  2gh  − 1 =  2gh  − 1
      v2 (2s / t)2
       
      α =  ght2  − 1
      2s2
       
      If you prefer to measure the angle of inclination of the ramp rather than its height you get a slightly different formula.
       
      α =  ght2  − 1  =  g(s sin θ)t2  − 1
      2s2 2s2
       
      α =  gt2sin θ  − 1
      2s
       
    2. The second question is best solved using Newton's laws of motion. The component of the weight parallel to the incline pulls the object down the incline while the frictional force pulls it up. Friction also exerts a torque that makes the object rotate about its center of mass. Pay special attention the α [alpha] symbols. Sometimes α means rotational acceleration and sometimes α is the coefficient of the moment of inertia. (The switch takes place between the second and third lines in the work shown below.)
                       
        translational   rotational
      ∑ F  =  m a   ∑ τ  =  I α
      W∥ − f  =  Ma∥   W∥R  =  Iα
      Mg sin θ − μMg cos θ  =  Ma∥   μMg(cos θ)R  =  (αMR2 ) (a∥ / R) 
      g sin θ − μg cos θ  =  a∥   μg cos θ  =  αa∥
       
      Divide these two equations to eliminate the acceleration parallel to the ramp and solve for the critical angle (or its tangent).
               
      g sin θ − μg cos θ  =  a∥
      μg cos θ αa∥
               
      tan θ  − 1  =  1  
      μ α  
               
      As the angle increases, friction decreases. Eventually the static friction force won't be strong enough to spin the object and it will slip. The critical angle at which this transition takes place is …
             
      tan θ = μ  ⎛
      ⎝      		 α + 1 ⎞
      ⎠
      α
             
      This formula is similar to one that was derived in an earlier part of this book (tan θ = μ). That formula was for an object on an incline that doesn't slip or roll.
  3. Determine the speed needed for a rigid wheel to roll over a rectangular step.
    • Answer it.
  4. Write something else.
    • Answer it.

conceptual

  1. basic, curtate, or prolate?
    1. pedal on a bicycle
    2. paddle on a steamboat
    3. ocean waves
    4. outermost point of a car tire
    5. outermost point of a train tire
    6. baton twirling

Resources

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