# Lecture mcmaster University

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```					             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
2
2
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
C
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
q
R

s = Rq
Physics 1D03
Quiz

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.
R
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
2
2

Rotational energy about CM axis: K rot  1 I
2
2

K  1 mvCM  1 I 2
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?

n
f
h

mg

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!) ...

y

N
f           a               x

mg

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
decrease?

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

Physics 1D03
d)
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
cylinder.
y

N
f          a           x

mg

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

Kinematics

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
Quiz

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
Summary
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
2
2
2

Rolling without slipping:
v=R
at = R 

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

Physics 1D03

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