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Chapter 2 The Laws of Motion, Part 2
September 14: Seesaws–Rotational motion
1
Observations about seesaws
• Equal-weight children balance a
seesaw.
• A balanced seesaw rocks back and
forth easily.
• Unequal-weight children don’t
normally balance the seesaw.
• Moving the heavier child inward
restores balance.
• Sitting closer to the pivot speeds up
the motion.
2
Translational motion and rotational motion
Translational motion: The overall movement of an
object from one place to another.
Rotational motion: Motion around a fixed point.
Examples of motion:
•A running train
•The spin of the earth
•The moving hands of your watch
•A wind turbine
•A flying stick
3
Question:
What keeps the merry-go-round
rotating?
Analogous question:
What keeps a skater moving?
Physics concept: Rotational Inertia
• A body that is rotating tends to remain rotating.
• A body that is not rotating tends to remain not
rotating.
4
Physical quantities on rotational motion:
• Angular position – an object’s orientation relative to a
certain line.
The SI unit of angular position is radian . 2p radian = 360°.
• Angular velocity – the change in angular position with
time. Describes how quickly the object is rotating.
angular speed 
change in angle
time
• Torque – A twist or spin.
These are the counterparts of position, velocity and force used in
describing translational motion.
5
More about angular velocity
•Angular velocity is a vector. It is
usually denoted as w.
•The direction of angular velocity is
defined by the right-hand rule:
Let the fingers of your right hand
curl in the way of the rotation, then
your thumb is pointing along the
angular velocity direction.
6
Question:
How much is the angular velocity
of the longest hand of my watch?
Answer:
angular speed 
change in angle 2π radian

 0.105 radian/sec ond
time
60 second
Question:
What is the direction of this angular velocity?
Answer:
The direction of the angular velocity is pointing into
the watch surface according to the right-hand-rule.
7
Newton’s first law of rotational motion
A rigid object that’s not wobbling and that is
free of outside torques rotates at a constant
angular velocity.
8
Question:
Andy Wang is trying to
harvest a bottle gourd. He
is not tall enough to reach
the stem. What can he do
now to easily pick up the
gourd?
9
Read: Ch2: 1
Homework: Ch2: E2
Due: September 25
10
September 16: Seesaws–Newton’s second law of
rotational motion
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Center of mass of an object
Center of mass: A point in or near an object about
which the mass is evenly distributed.
•For a symmetric object the center of mass is located at its
geometric center.
•For a less symmetric object the location of the center of
mass depends on its mass distribution.
•Center of mass is very useful in describing the motion of
an object.
12
Examples of center of mass
13
Simplification of motion:
A free object naturally spins about its center of mass (e.g., the
earth).
The motion of an object can be decomposed into
1)a translational motion of its center of mass, and
2)a rotational motion around its center of mass.
Example: A diver.
Demo: Flying wood board.
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More physical quantities on rotational motion:
• Rotational mass (Moment of inertia) – measure of the
rotational inertia of an object. The SI unit of rotational mass
is kilogram·meter2.
• Angular acceleration – the change in angular velocity with
time. Describes how quickly the angular velocity is changing.
The SI unit of angular acceleration is radian/second2.
angular accelerati on 
change in angular ve locity
time
They are the counterparts of mass and acceleration used in
describing translational motion.
15
More about rotational mass:
• Rotational mass of an object measures how
difficult to change its angular velocity. The
value of rotational mass depends on the
ordinary mass of the object and its distribution
around the specific rotational axis.
• The contribution of each portion of mass of an
object to the rotational mass of the object is
proportional to its distance squared from the
rotational axis:
I  m1d12  m2d 22  m3d 32  
For the tennis racket, which case has the
largest rotational mass?
16
More about torque
• The distance from the rotational axis to
where the force is exerted is called the
lever arm.
• Only the component of force
perpendicular to the level arm contributes
to torque. Think on how you open a door.
• The direction of torque is determined by
the right-hand-rule.
• torque  lever arm
 force perpendicu lar to the lever arm
  r  F
17
Newton’s second law of rotational motion
The angular acceleration of an object is equal to the
net torque exerted on it divided by its rotational
mass. The angular acceleration is in the same
direction as the net torque.
net torque
angular accelerati on 
rotational mass
net torque  rotational mass  angular accelerati on
 net  I  
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Read: Ch2: 1
Homework: Ch2: E6,12; P1,6
Due: September 25
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September 18: Seesaws–Rotational work
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Balancing a seesaw
A balanced seesaw (with riders) is one that experiences no
net torque due to gravity, so that it can rotate smoothly.
When calculating torque, the gravity of an object can be
thought as being exerted at the center of gravity of the object.
For smooth rotation the torque
caused by the gravity of the seesaw
must be zero. The center of gravity
of the seesaw must be at the pivot.
Two children with different weights
can balance the seesaw by sitting at
different distances from the pivot.
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Mechanical advantage of a lever
For a balanced seesaw:
(a) If the two weights are equal, the
work that the descending child
does on the board equals the
work that the board does on the
rising child.
(b) If one child is heavier than the
other, the two works involve
different forces over different
distances, but they are still equal.
(c) Therefore the board only
transfers energy from one child
to the other.
22
Question:
The boy exerts a torque on the seesaw board. Does the board
also exert a torque on the boy?
Answer:
Yes. The board exerts an equal but oppositely directed torque
on the boy. The net torque on the boy is zero, so he rotates
smoothly.
23
Newton’s third law of rotational motion
For every torque one object exerts on a second
object, there is an equal but oppositely directed
torque that the second object exerts on the first
object (provided that the two objects rotate about the
same axis).
Examples:
•Wind turbine and generator.
• A child running on a merry-go-round.
• Spanner wrench and nut.
24
Work done in rotation
work = force · distance along the force direction
work done in a full revolution = force · distance
= force · (level arm · 2p )
= (force · level arm) · 2p
= torque · 2p
Work done in any amount of rotation:
work = torque · angle (in radian)
W   
25
Question:
The wind exerts a torque of 500 Newton·meter on a wind
turbine. The turbine rotates 2 cycles per second. How much
work does the wind do on the turbine per second?
Answer:
work = torque · angle (in radian)
 500 N  m  2  2p
 6280 J
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Read: Ch2: 1
Homework: Ch2: E15
Due: September 25
27
September 21: Wheels–Friction
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Demo: Pushing a textbook on a table.
Question: Why does the book move slowly even though
I push it continuously? (Newton’s second law says that it
should accelerate…).
Answer: There exist a frictional force between the book
and the table. The frictional force is always along the
surfaces and opposing the sliding motion.
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More about frictional force:
1)Its strength depends on how hard the two surfaces are
pressed against one another.
2)Its strength depends on how slippery the two surfaces
are.
3)Its strength depends on whether or not the two surfaces
are moving relative to one another.
4)Its strength does not depend much on the area of
contact between the surfaces.
5)It adjusts itself in response to the situation.
6)Newton’s third law of motion applies.
7)Friction is ubiquitous. It can help us. It can bother us.
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A microscopic view of friction
Surfaces have microscopic hills and valleys.
When two surfaces are relatively moving, the
structures collide with each other and produce
a horizontal force.
Frictional forces are actually resulted from the
electromagnetic forces between the structures.
Demo: Pulling textbooks on a table.
The magnitude of the frictional force is proportional to
the force pressing the two surfaces (normal force):
Ffriction    N
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The two types of friction
1.Static friction
• Acts to prevent objects from starting to slide.
• Ranges from zero to an upper limit.
2.Sliding friction
• Acts to stop objects that are already sliding.
• Has a fixed magnitude.
(Maximum) static frictional force > sliding frictional force
•In static friction surface features can interpenetrate better.
•Frictional force drops when sliding begins.
32
Question:
The signal light turns green and you’re in a hurry. Will
your car accelerate faster if you
1) skid your wheels and “burn rubber” or
2) just barely avoid skidding your wheels?
Answer:
Just barely avoid skidding your wheels.
Question:
Why is it dangerous (sometimes extremely dangerous) to
abruptly step on the brake of your car in a snowy day?
33
Friction and energy
1.Static friction
• There is no relative motion between the surfaces.
• No work is done and so there is no wear.
2.Sliding friction
• The two surfaces are relatively moving.
• Some work “disappears” and becomes thermal
energy.
• The surfaces experience wear.
34
Physical quantity: Power
Power is the work done in a unit time. It measures how fast an
object is doing work. The SI unit of power is Joule-persecond, called watt (W).
power 
1 kilowatt (kW) = 1000 watt
1 horsepower = 745.7 watt
P
work
time
W
t
Power in translational motion:
power = force · distance /time = force · velocity
Power in rotational motion:
power = torque · angle /time = torque · angular velocity
35
Read: Ch2: 2
Homework: Ch2: E22,27,29
Due: October 2
36
September 23: Wheels–Kinetic energy
37
The many forms of energy
1.Kinetic energy: Energy of an object because of motion.
2.Potential energy: Energy stored in objects because of
their positions, shapes and structures.
•
•
•
•
•
•
Gravitational potential energy
Elastic potential energy
Electric potential energy
Chemical potential energy
Magnetic potential energy
Nuclear potential energy
3. Thermal energy: A disordered mixture of kinetic
and potential energy at molecular level.
38
Eliminating sliding frictions
Rollers:
•Rollers eliminate sliding friction
at roadway.
•Rollers keep moving out from
under the object.
•Rollers are not so convenient.
39
Wheels:
•Eliminate sliding friction at roadway.
•Convenient because they don’t pop out.
Question: Who is turning the wheels when the cart is accelerating?
Static friction exerts torques on the wheels.
Question: Why do I still need to pull the cart even if it is moving at
a constant velocity?
Wheel hubs have sliding friction with the axels.
40
Bearings:
•Eliminate sliding friction in wheel hubs.
•Behave like automatically recycling rollers.
41
Practical wheels:
•When accelerating, free
(non-driving) wheels are
turned by a backward static
fraction.
•When accelerating,
powered (driving) wheels
are turned by the engine and
a forward static friction.
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Kinetic Energy
1. A translationally moving object has a kinetic energy of:
1
kinetic energy  · mass · speed 2
2
1
K   m  v2
2
2. A rotationally moving object has a kinetic energy of:
1
· rotational mass · angular speed 2
2
1
K   I w 2
2
kinetic energy 
43
Question:
My car is running on a street in Macomb with a speed of
30 mile/hour. You have a similar car which is running on
highway 88 at 60 mile/hour.
How much more kinetic energy does your car have
compared to mine?
Answer:
Because the kinetic energy is proportional to the
square of speed, your car has four times kinetic energy
compared to mine.
44
Read: Ch2: 2
No homework
45
September 25: Bumper Cars–Linear momentum
46
Momentum: A basic physical quantity
• Momentum (linear momentum) measures the
translational motion of an object. Specifically
momentum  mass  velocity
p  m v
• The SI unit of momentum is kilogram·meter/second.
• Momentum is a vector quantity. It is in the direction
of the velocity. On a certain direction, it can be
positive or negative.
47
Question:
My car is running on a street in Macomb with a speed of
30 mile/hour. You have a similar car which is running on
highway 88 at 60 mile/hour.
How much more momentum does your car have
compared to mine?
Answer:
Because the magnitude of momentum is proportional to
the speed, your car has two times momentum compared
to mine.
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Impulse
• Momentum is transferred by impulse. Specifically
impulse  force  time
impulse  F  t
• The SI unit of impulse is newton·second.
• Impulse is a vector quantity. It is in the direction of
the force.
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The impulse-momentum relationship
• The change in the momentum of an object equals
the net impulse exerted on the object:
Change in momentum  impulse received
p  F  t
• It comes from Newton’s second law of motion:
v
F  ma  m
t
 p  m  v  Ft
It is therefore correct!
50
Question: My car has a mass of 2500 kg and it is
running at a velocity of 4 m/s. It is out of control because
the brake is defect. How much impulse is needed to stop
my car?
Answer:
impulse needed  change in momentum
 m  v
 2500 kg  4 m/s
 10,000 kg  m/s
 10,000 N  s
51
Question: I am sliding on ice. My mass is 75 kg and my
velocity is 2 m/s. How much is my momentum?
Answer:
momentum  mass  velocity  75 kg  2 m/s  150 kg  m/s.
Question: How much impulse should I receive so that I
can stop?
Answer:
150 kg  m/s, or 150 N  s.
Question: If I push the ice forward with a force of 10 N,
how long time does it take me to stop?
Answer: impulse  force  time
time 
impulse 150 N  s

 15 s.
force
10 N
52
Read: Ch2: 3
Homework: Ch2: P9,10,11
Due: October 2
53
September 28: Bumper Cars–Momentum
conservation
54
Review: Momentum, impulse and their relations
• Momentum measures the translational motion of an
object:
momentum  mass  velocity
p  m v
• Impulse transfers momentum:
impulse  force  time
impulse  F  t
• Impulse-momentum relationship: The change in the
momentum of an object equals the net impulse
received by the object:
Change in momentum  impulse received
p  F  t
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Demo: Tablecloth revisited (impulse-momentum view)
• Goal: Let v of the glasses be as small as possible.
• Principle: Change in the momentum of the glasses equals
to the impulse received:
F  t
m  v  F  t  v 
m
• Known: F (sliding frictional force) is small and fixed for
given glasses and cloth. The mass of the glasses m is also
fixed.
• Solution: Keep t as short as possible.
56
Law of Conservation of Momentum
(A fundamental law in physics)
When there is no external force, the momentum
of a system remains unchanged.
Movie: Space experiment: momentum conservation
Discussion: Newton’s third law of motion
57
Demo: Momentum conservation in Collision
(Newton’s pendulum).
58
Examples of momentum conservation
59
Question:
Your are caught by a monster and put onto a frozen river
covered by slippery ice which cannot provide any frictional
force. The monster knows a little physics and believes that
without a frictional force you cannot let the ice help you to
slide, which is unfortunately true.
How can you manage to escape?
60
Read: Ch2: 3
Homework: Ch2: P13
Due: October 9
61
September 30: Bumper Cars–Angular momentum
62
Angular momentum: A basic physical quantity
• Angular momentum measures the rotational
motion of an object. Specifically
angular momentum  rotational mass  angular ve locity
L  I ω
• The SI unit of angular momentum is
kilogram·meter2/second.
• Angular momentum is a vector quantity. It is in the
direction of the angular velocity.
63
Angular impulse
• Angular momentum is transferred by angular
impulse. Specifically
angular impulse  torque  time
 τ t
• The SI unit of angular impulse is
newton·meter·second.
• Angular impulse is a vector quantity. It is in the
direction of the torque.
64
The angular impulse-angular momentum relationship
• The change in the angular momentum of an object
equals the net angular impulse exerted on the
object:
Change in angular momentum  angular impulse received
L  τ  t
• It comes from Newton’s second law of rotational
motion:
ω
τ  I α  I 
t
 L  I  ω  τ  t
It is therefore correct!
65
Law of Conservation of Angular Momentum
(A fundamental law in physics)
When there is no external torque, the angular
momentum of a system remains unchanged.
66
Question:
You are riding on the edge of a spinning playground
merry-go-round. If you pull yourself to the center of the
merry-go-round, what will happen to its rotation?
Answer:
The total angular momentum is conserved. The
rotational mass is reduced when you go to the center of
the merry-go-round. Therefore it will spin faster.
67
Examples of angular momentum conservation
68
Demo: Angular momentum conservation
69
70
Read: Ch2: 3
Homework: Ch2: E38;P14
Due: October 9
71