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PreClass Notes: Chapter 10, Sections 10.4,10.5
• From Essential University Physics 3rd Edition
• by Richard Wolfson, Middlebury College
• ©2016 by Pearson Education, Inc.
• Narration and extra little notes by Jason Harlow,
University of Toronto
• This video is meant for University of Toronto
students taking PHY131.
Outline
“Our description of rolling motion
leads to a point you may at first find
absurd: In a rolling wheel, the point
in contact with the ground is,
instantaneously, at rest!” –
R.Wolfson
• 10.4 Rotational Energy
• 10.5 Rolling Without
Slipping
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© 2012 Pearson Education, Inc. Slide 1-3
Rotational Kinetic Energy
• A rotating object has kinetic energy associated
with its rotational motion. K
rot 1
2I 2
Table 10.1
© 2012 Pearson Education, Inc. Slide 1-4
Work-Kinetic Energy Theorem for Rotation
• The change in an object’s rotational kinetic energy is equal to
the net work done on the object by torques.
• Definition of net work for torques:
𝑊rot = 𝜃1
𝜃2
𝜏net𝑑𝜃
𝑊rot =12𝐼𝜔2
2 − 12𝐼𝜔1
2
• Work-Kinetic Energy Theorem:
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Rolling without slipping
S' frame: the axle
S frame: the ground
right theto ,vV
ω
The wheel rotates with angular speed ω.
The tangential speed of a point on the rim is v = ωR,
relative to the axle.
In “rolling without slipping”, the axle moves at
speed v. This is the S' frame.
Rolling without slipping
S' frame: the axlevv 1
vv 3
vv 4
right theto ,vV
vv 2
V = ωR is the speed of the S' frame
relative to the ground.
v = ωR is the
tangential speed
of any point on
the rim.2
1
4
3
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Rolling without slipping
1v
V
Vvv
1v
Point 1: Top of the wheel
RvvvVvv 2211
In S frame (the ground), the top point
moves at speed 2v = 2ωR
S frame: the ground
Rolling without slipping
2v
V
Vvv
2v
Point 2: right-side of the wheel
RvvvVvv 2222
22
In S frame (the ground), the right side of
the wheel is moving on a diagonal down
and to the right.
S frame: the ground
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Rolling without slipping
3v
V
Vvv
03 v
Point 3: Bottom of the wheel
033 vvVvv
In S frame (the ground), the bottom point
is at rest.
S frame: the ground
Rolling without slipping
S' frame: the axle S frame: the ground
vv 1
vv 22
vv 3
vv 4
right theto ,vV
Vvv
vv 21
vv 24
03 v
vv 2
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Rolling without slipping
right theto ,RV
ω
The wheel rotates with angular speed ω.
Since the bottom point is always at rest, it is
static friction which acts between the ground and
the wheel.
The axle moves with linear speed v = ωR,
where R is the radius of the wheel.
Rolling Without Slipping
3 sides: bottom pivot point does not move: fixed point.
Another way to look at it…
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Rolling Without Slipping
4 sides: bottom pivot point does not move: fixed point.
Another way to look at it…
Rolling Without Slipping
8 sides: bottom pivot point does not move: fixed point.
….etc, etc.
If you have an infinite number of sides (circle), the
bottom pivot point still should not move.
Another way to look at it…
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Got it?
A car accelerates away from a stop-sign.
What is the main external force on the car which
provides the net force which causes it to accelerate?
A. Gravity
B. Kinetic friction
C. Normal Force
D. Static friction
E. Thrust
Rolling Without Slipping Constraint
Center of
mass motion
of rolling
circle
Rotational
motion
∆𝑥 = ∆𝜃 ∙ 𝑅
𝑣 = 𝜔 ∙ 𝑅
𝑎 = 𝛼 ∙ 𝑅
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Example 10.12 Energy Conservation: Rolling
Downhill
A solid ball of mass M and radius R starts from rest and rolls down a hill.
Its center of mass drops a total distance h. Find the ball’s speed at the
bottom of the hill.
𝐾𝑡𝑟𝑎𝑛𝑠 =12𝑀𝑣2
𝐾𝑟𝑜𝑡 =12𝐼𝜔2
𝑈 = 𝑀𝑔ℎ
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Got it?
• A hollow ring and a solid disk roll without slipping down an
inclined plane. Which reaches the bottom of the incline first?
A. The solid disk reaches the
bottom first.
B. The hollow ring reaches the
bottom first.
C. Both balls reach the bottom at
the same time.