Transcript From last time… - University of Wisconsin–Madison

From last time
• Position, velocity, and acceleration
– velocity = time rate of change of position
– acceleration = time rate of change of velocity
– Particularly useful concepts when
• velocity is constant (undisturbed motion)
• acceleration is constant (free falling object)
HW#1:
Due
HW#2:
Chapter 3: Conceptual: # 2, 26, 30, 40
Chapter 3: Problems: # 4, 6, 17
Physics 107, Fall 2006
1
Galileo Uniform acceleration from rest
7
Acceleration = const = a = 9.8
Velocity
m/s2
VELOCITY ( m/s)
6
= (acceleration)x(time)
= at
3
2
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
Falling
TIME (Ball
s)
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
TIME ( seconds )2
3
2.5
DISTANCE ( meters )
= (average vel)x(time)
= (1/2)at x t = (1/2)at2
4
1
Uniformly increasing velocity
Distance
5
2
1.5
1
0.5
0
Physics 107, Fall 2006
Questions
A car slows down from 60mph to 0 in 6 seconds
How far does the car go during that time?
A. 0.1 mile
B. 0.2 mile
C. 0.05 mile
Since speed changes uniformly with time
(from 60 mph to 0 mph), so average speed is 30 mph.
Distance = average speed x time
= (30 miles/hour) x (6 seconds) =
= (30 miles/hour)
x 2006
(1/600 hr) = 1/20 mile
Physics 107, Fall
3
Falling object: constant acceleration
• Falling objects have constant acceleration.
• This is called the acceleration of gravity
9.8 m/s/s = 9.8 m/s2
• But why does gravity result in a constant
acceleration?
• Why is this acceleration independent of mass?
Physics 107, Fall 2006
4
Tough questions
• These are difficult questions.
Maybe not completely answered even now.
• But tied into a more basic question:
– What causes acceleration?
– Or, how do we get an object to move?
A hot topic in the 17th century.
Descartes was a major player in this.
Physics 107, Fall 2006
5
Descartes’ view…
• Motion and rest are primitive states of a body
without need of further explanation.
• Bodies only change their state when acted upon
by an external cause. This is similar our
concept of inertia
That a body, upon coming in contact with a
stronger one, loses none of its motion; but
that, upon coming in contact with a weaker
one, it loses as much as it transfers to that
weaker body Momentum and it’s conservation
Physics 107, Fall 2006
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Inertia
• Principle of inertia: object continues at
constant velocity unless disturbed.
– Need a disturbance to change the velocity.
• Inertia measures the degree to which an object
at rest will stay at rest.
– Objects with lots of inertia don’t change motion as
much as lighter objects subject to the same
disturbance.
– They are more difficult to accelerate
Physics 107, Fall 2006
7
Quantifying Inertia: Mass and
Momentum
• Same disturbance applied to different objects
results in different velocities
(e.g. hitting bowling ball and golf ball w/golf club).
• But the product
mass  velocity
is the same
(e.g. for the bowling ball and the golf ball).
• Momentum = (mass)(velocity)
Physics 107, Fall 2006
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Momentum conservation
• Can easily describe interactions of objects.
• The total momentum
(sum of momenta of each object)
of the system is always the same.
• We say that momentum is conserved.
– Between the golf ball and the golf club
• Momentum can be transferred from one object
to the other, but it does not disappear.
Physics 107, Fall 2006
9
Momentum conservation: equal
masses
1
m = 1 kg
v = 1 m/s
p = mv = 1 kg-m/s
After collision:
Total momentum =
0 kg-m/s+1 kg-m/s = 1 kg-m/s
2
m =1 kg
v = 0 m/s
p=0
Before collision:
Total momentum =
1 kg-m/s+0 kg-m/s = 1 kg-m/s
1
m =1 kg
v = 0 m/s
p =Fall
0 kg-m/s
Physics 107,
2006
2
m = 1 kg
v = 1 m/s
p = 1 kg-m/s10
Another possibility
1
m = 1 kg
v = 1 m/s
p = mv = 1 kg-m/s
2
m =1 kg
v = 0 m/s
p=0
Before collision:
Total momentum before =
1 kg-m/s+0 kg-m/s = 1 kg-m/s
After collision: balls stick together
1
Total momentum after = 1 kg-m/s
Physics 107, Fall 2006
2
m = 1 kg + 1 kg = 2 kg
v = 0.5 m/s
p = 1 kg-m/s
11
Question
Two ice skaters, Joe and Jane, are initially at rest.
Joe is more massive than Jane. They push off of
each other. Compared to Jane, Joe is moving
A. faster
B. slower
C. the same.
The momentum before they push off is zero, so it
also is after they push off. The momenta must be
equal and opposite. Since Joe is more massive, and
Physics 107, Fall 2006
momentum = mass x velocity,
his speed is slower.12
What about Newton?
• Like Galileo and Descartes, Newton has a law of
inertia.
• Newton’s first law:
Every body perseveres in its state of rest, or of uniform
motion in a right line, unless it is compelled to change
that state by forces impressed upon it.
• The ‘force’ is the ‘external disturbance’ of
Galileo and Descartes
Physics 107, Fall 2006
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Newtonian Forces
• Newton made a definition of force that described
how momentum was transferred.
• He viewed it as a continuous process rather than
the immediate transfer of Descartes and Galileo.
• This makes a connection with our intuitive
understanding of ‘force’ as a push or a pull.
Physics 107, Fall 2006
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Newton’s second law
• The change in motion is proportional to the
motive force impressed; and is made in the
direction of the right line in which that force is
impressed.
(Momentum change)=(Applied force)(time interval)
Change in momentum = p
Applied force = F
Time interval = t
Physics 107, Fall 2006
p = F t
15
Impulse
• The momentum change is called an impulse
when it occurs over a very short time.
• An impulse is a short ‘disturbance’
exerted on an object.
• It is equal to the momentum change of the object.
• It’s units are the same as that of momentum
– Units are kg-m/s
• Makes a connection between Descarte and Newton.
Physics 107, Fall 2006
16
(change in momentum)=(Applied force)(change in time)
change in momentum

 applied force
change in time
Momentum = (mass)  (velocity)
Change in momentum = (mass)  (change in velocity)

change in velocity
 mass 
 applied force
change in time
acceleration
 Newton’s
F = ma
Force = (mass)  (acceleration)
second law
Physics 107, Fall 2006
force
acceleration = 17
mass
Force results in acceleration
• A body will accelerate (change its velocity)
when another body exerts a force on it.
• This is also a change in momentum.
• But what is a force?
– Push
– Pull
– Jet thrust
F1
Physics 107, Fall 2006
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More than one force…
• Total force determines acceleration
• If F1 and F2 balance, acceleration is zero.
F1
F2
Physics 107, Fall 2006
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Back to falling bodies
• A free-falling body moves with constant
acceleration.
• Newton says that this means there is a constant
force on the falling body.
• This is the gravitational force,
and is directed downward.
Physics 107, Fall 2006
20
Question
When the vectron hovers near the ceiling, the
propeller force compared to hovering near the
floor is
A. Greater.
B. Less.
C. The same.
Gravity exerts a force downward. When the vectron
hovers, its velocity is constant, so the acceleration is
zero. This means the net force is zero. The propeller
Physics 107, Fall 2006 force
21
force balances the gravitational
Types of forces
Physics 107, Fall 2006
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The Four Forces
Strong nuclear force
Electromagnetic force
Weak nuclear force
Gravity
Decreasing strength
1.
2.
3.
4.
– Only gravity and electromagnetic forces
are relevant in classical mechanics
( motion of macroscopic objects ).
Physics 107, Fall 2006
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Force and acceleration
• Larger force gives larger acceleration
• Directly proportional:
aF
• But clearly different bodies accelerate
differently under the same force.
– Heavier objects
 are harder to push.
– Proportionality constant may depend on weight?
Physics 107, Fall 2006
24
Inertia again
• But we already said that inertia characterizes
a body’s tendency to retain its motion
(I.e. to not change its velocity),
We say a heavier object has more inertia.
• But inertia and weight are different
– A body in space is weightless,
but it still resists a push
Physics 107, Fall 2006
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Mass
• Define mass to be
‘the amount of inertia of an object’.
• Can also say mass characterizes the amount of
matter in an object.
• Symbol for mass usually m
• Unit of mass is the kilogram (kg).
• Said before that
aF
• Find experimentally that

Physics 107, Fall 2006

Force
Acceleration=
Mass
F
a
m26
Force, weight, and mass
F  ma F  (kg)(m/s 2 )
 kg  m /s  Newton
2
• 1 Newton = force required to accelerate a
1 kg mass at 1 m/s2.
But then what is weight?
—Weight is a force, measured in Newton’s
—It is the net force of gravity on a body.
—F=mg, g=F/m
Physics 107, Fall 2006
27
What do you think?
Suppose you are on the moon
instead of on earth
A. Your weight is less but your mass is the same.
B. Both your weight and mass are less than on earth.
C. Your weight is less and your mass is zero.
Mass is an intrinsic characteristic of a body.
The force of gravity on the body (weight) will
depend on the other
Physicsbodies
107, Fall 2006around it.
28
Is ‘pounds’ really weight?
• In the English system (feet, pounds, seconds), pounds
are a measure of force.
• So it is correct to say my weight is 170 pounds.
• Then what is my mass?
F 170lbs
m 
 5.3 slugs!!
2
g 32 ft / s

Physics 107, Fall 2006
29
Momentum conservation
• We said before that an impressed force changes
the momentum of an object.
• We also said that momentum is conserved.
• This means the momentum of the object applying
the force must have decreased.
• According to Newton, there must be some force
acting on that object to cause the momentum
change.
Physics 107, Fall 2006
30
Newton’s third law
• This is the basis for Newton’s third law:
To every action there is always opposed an
equal reaction.
This is momentum conservation in the
language of forces.
Physics 107, Fall 2006
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