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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.