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Lecture mcmaster University


									             Rolling Motion

• Combined translational and rotational motion
• “Rolling without slipping”
• Dynamics of rolling motion

           Serway & Jewett: 10.9

                                      Physics 1D03
        General Motion of a Rigid Body
   F  ma     CM
                        Gives linear acceleration of the position of the
                        center of mass.

   CM  ICM          Gives angular acceleration of the body about
                        the center of mass.

  Kinetic energy:    K  K trans  K rot  1 mvCM  1 I CM  2

These are special properties of the centre of mass only

(although   I is also true for any stationary axis of rotation).

                                                           Physics 1D03
                       Rolling Motion

Different points on a rolling object
have different velocities (blue
vectors). The velocity of each
particle (blue) is the velocity of the
centre (red) plus the velocity r                     vC
relative to the centre (green):

v = vCM + rω

The point in contact with the ground is momentarily
stationary. The point on the top of the wheel moves
forward at twice the speed of the centre.

                                                           Physics 1D03
If the body rolls without slipping, then its angular velocity and
angular acceleration are related to the linear velocity and tangential
acceleration of the centre:

               v=R       and      a=R

Proof: When the object turns through an angle q, it moves forward
a distance s = Rq (if it doesn’t slip).

                                  s = Rq ; take derivatives wrt t to get:
                                          vcentre = R
                                   and    acentre = R

               s = Rq
                                                         Physics 1D03

There is friction between a wheel and the road,
otherwise a car could not accelerate. This friction is
what causes a rolling wheel to stop.

         a) True             b) False

                                            Physics 1D03
Friction and Rolling Resistance

Friction is necessary to create a torque                                  pull
which can cause rotation and rolling.
This is static friction if the wheel        f
rolls without slipping. At the point
                                                                       Rf
of contact between the road and
wheel, there is no relative motion,
and friction does no work.

“Rolling resistance” which slows the wheel results from the tire
(and the road surface) flexing inelastically at the contact point.
This can be a small effect, so wheels roll a long way.

                                                           Physics 1D03
Symmetrical rolling objects (CM in centre):

1. Dynamics: Use aCM  R                             these are related by
                                                      the condition for rolling
 CM motion:     F  maCM
 Rotation about CM:       I

2. Energy:       Use vCM  R
 Translational energy from CM: K trans  1 mvCM

 Rotational energy about CM axis: K rot  1 I

                 K  1 mvCM  1 I 2
                     2        2
                                              These are related by
                                              the condition for rolling

                                                             Physics 1D03
    Example: A cylinder (mass m, radius R) rolls down a ramp
    without slipping. What is its speed after it has descended a
    vertical height h?



Which forces do work?

                                                             Physics 1D03
How would you calculate the force of static friction?

                 forces (x components): mg sin q – f = maCM

                 torques (clockwise):        Rf = ICM 
                              rolling:       aCM= R
                                                       mg sin q
                                              f 
                                                    1  mR 2 I CM 
      solve (exercise for student!) ...


             f           a               x


                                                                Physics 1D03
Example: A solid cylinder and a
hollow pipe roll down a ramp
without slipping.

a) Which gets to the bottom first?

b) If they have equal masses,
which has the greatest kinetic
energy at the bottom?

c) If the surface were slippery, would the time increase or

d) For a moderate coefficient of friction, which can roll down the
steepest slope without slipping?

                                                              Physics 1D03
            mg sin q
     f                         Icyl = ½ mR2
         1  mR 2 I CM        Ipipe = mR2

                                So f is larger if I is larger.

                                So the pipe needs more
                                friction than the solid

     f          a           x


                                                     Physics 1D03
The general motion of a rigid body (i.e., not pure rotation about a fixed
axis) can be described as translation of the center of mass, plus
rotation about the center of mass.

                                                  v y t    vCM

     rCM t                                         CM
                     Refer to motion of
     v CM t        center of mass                               vx t 
     a CM t 
                                                             aCM = g

                              Refer to rotation about an axis though
   q t ,  t ,  t     the center of mass, which travels with the
                              center of mass

                                                              Physics 1D03

On a dry day, a round steel pipe is released from rest in a concrete
dainage ditch. It rolls down without slipping, and then rolls up the other
side to almost its original height.

On a rainy day, there is less friction, and the pipe slips partially on the
way down before it rolls. How far up the other side will it go?

 A) Just as high as on the dry day.
 B) A little higher.
 C) Not as high.

                                                               Physics 1D03
For arbitrary motion of a rigid body, divide the motion into a linear
motion of the centre of mass, plus rotation about the centre of
mass. Then:
                    F  ma    CM

                     I
                        CM     CM   
                   K  K trans  K rot  1 mvCM  1 I CM  2

 Rolling without slipping:
                             at = R 

    Practice problems: 11-5, 11-47, 11-63

                                                            Physics 1D03

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