Relative Velocity
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Two observers moving relative to each other
generally do not agree on the outcome of an
experiment
However, the observations seen by each are related
to one another
A frame of reference can described by a Cartesian
coordinate system for which an observer is at rest
with respect to the origin
Different Measurements,
example
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Observer A measures
point P at +5 m from
the origin
Observer B measures
point P at +10 m from
the origin
The difference is due to
the different frames of
reference being used
Different Measurements,
another example
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The man is walking on the
moving beltway
The woman on the beltway
sees the man walking at his
normal walking speed
The stationary woman sees
the man walking at a much
higher speed
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The combination of the
speed of the beltway and
the walking
The difference is due to the
relative velocity of their
frames of reference
Relative Velocity, generalized
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Reference frame SA is
stationary
Reference frame SB is
moving to the right
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relative to SA at v AB
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This also means
that SA
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moves at – vBA relative to
SB
Define time t = 0 as that
time when the origins
coincide
Notation
The first subscript represents what is being
observed
l  The second subscript represents who is
doing the observing
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l  Example v AB
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The velocity of A as measured by observer B
Relative Velocity, equations
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The positions as seen from the two reference frames
are related through the velocity
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l  r
= rPB + vBAt
PA
The derivative of the position equation will give the
velocity equation
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l  uPA = uPB + v BA
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uPA is the velocity of the particle P measured by observer A
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uPB is the velocity of the particle P measured by observer B
These are called the Galilean transformation
equations
Special Relativity velocity addition (
):
Acceleration in Different
Frames of Reference
The derivative of the velocity equation will
give the acceleration equation
l  The acceleration of the particle measured by
an observer in one frame of reference is the
same as that measured by any other
observer moving at a constant velocity
relative to the first frame.
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The Laws of Motion
Sir Isaac Newton
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25 Dec 1642 – 31 Mar
1727* (84 years old)
Formulated basic laws
of mechanics
Discovered Law of
Universal Gravitation
Invented calculus
Many observations
dealing with light and
optics
* Add 11 days for modern calendar
Engraving made from
1726 painting by John
Vanderbank
Newton aged 46. Earliest
portrait, painted in 1689 by
Godfrey Kneller
Death mask 1727
Aged 84
Woolsthorpe Manor and apple tree 1820
1666 notes by John Conduit, first
mention of apple and gravity!
(when Newton was 24)
“He first thought of his system of
gravitation which he hit upon by
observing an apple fall from a tree”
Woolsthorpe Manor and apple tree 1998
Newton performing his prism experiment. Painting by Sascha Grusche (2015)
Newton’s bedroom at
Woolsthorpe Mannor today
Physics World poll 1999 - Who is the greatest physicist ever?
Some of Isaac Newton’s sins, he listed himself
in 1662, aged 19
“Neglecting to pray”
“Falling out with the servants”
“Punching my sister”
“Making a mousetrap on a Sunday morning”
“Eating an apple in church”
“Robbing my mother’s box of plums and sugar”
“Peevishness with my mother” (peevish = being angry)
“Swimming in a kimnel on Thy day” (kimnel=a vessel for water, e.g. a trough)
“Calling Dorothy Rose a jade” (disreputable or bad-tempered)
“Denying a crossbow to my mother and grandmother although I knew of it”
“Peevishness over a piece of bread and butter at Mr Clark’s”
“Stealing cherry cobs from Edward Storer” (cob=piece of wood? Building
material? For some experiment he was doing?)
“Beating Arthur Storer”
“Putting a pin in John Keys hat to prick him”
“Threatening my father and mother to burne them and the house over them”
“Having unclean thoughts”
Force
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Forces are what cause any change in the
velocity of an object
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Newton’s definition
A force is that which causes an acceleration
Classes of Forces
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Contact forces involve physical contact
between two objects
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Examples a, b, c
Field forces act through empty space
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No physical contact is required
Examples d, e, f
Fundamental Forces
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Gravitational force
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Electromagnetic forces *
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Between subatomic particles
Weak forces
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Between electric charges
Nuclear force
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Between objects
Arise in certain radioactive decay processes
Note: These are all field forces
* Barrow & Webb:
http://www.scientificamerican.com/article.cfm?id=inconstant-constants&ref=sciam
More About Forces
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A spring can be used to calibrate the magnitude of a
force
Doubling the force causes double the reading on the
spring
When both forces are applied, the reading is three
times the initial reading
Vector Nature of Forces
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The forces are applied
perpendicularly to each
other
The resultant (or net)
force is the hypotenuse
Forces are vectors, so
you must use the rules
for vector addition to
find the net force acting
on an object
Newton’s First Law
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If an object does not interact with other
objects, it is possible to identify a reference
frame in which the object has zero
acceleration
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This is also called the law of inertia
It defines a special set of reference frames called
inertial frames
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We call this an inertial frame of reference
Inertial Frames
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Any reference frame that moves with constant
velocity relative to an inertial frame is itself an inertial
frame
A reference frame that moves with constant velocity
relative to the distant stars is the best approximation
of an inertial frame
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We can consider the Earth to be such an inertial frame,
although it has a small centripetal acceleration associated
with its motion
Newton’s First Law –
Alternative Statement
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In the absence of external forces, when viewed from
an inertial reference frame*, an object at rest
remains at rest and an object in motion continues in
motion with a constant velocity
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Newton’s First Law describes what happens in the absence
of a force
l  Does not describe zero net force
Also tells us that when no force acts on an object, the
acceleration of the object is zero
* A non-accelerating reference frame
Inertia and Mass
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The tendency of an object to resist any attempt to
change its velocity is called inertia
Mass is that property of an object that quantifies
how much resistance an object exhibits to changes
in its velocity
Masses can be related to the accelerations
produced by a given force acting on them:
m1 a2
=
m2 a1
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The magnitude of the acceleration acting on an object is
inversely proportional to its mass
More About Mass
Mass is an inherent property of an object
l  Mass is independent of the object’s
surroundings
l  Mass is independent of the method used to
measure it
l  Mass is a scalar quantity
l  The SI unit of mass is kg
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Mass vs. Weight
Mass and weight are two different quantities
l  Weight is equal to the magnitude of the
gravitational force exerted on the object
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Weight will vary with location
Example:
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wearth = 180 lb; wmoon ~ 30 lb
mearth = 2 kg; mmoon = 2 kg
What is mass?
“Gravitational mass”, a
“characteristic constant” of
the particles experiencing
a gravitational force
Gravitational force
Inertial force
Gmg1mg 2
F=
2
r
F = ma
€
DOES
€
“Inertial mass”, a different
“characteristic constant” of
the accelerated particle
GRAVITATIONAL MASS
=1 ?
INTERTIAL MASS
Answer: “Yes” macroscopically; “Maybe not” quantum mechanically!
http://www.technologyreview.com/view/419367/new-quantum-theory-separates-gravitational-and-inertial-mass/
€
Newton’s Second Law
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When viewed from an inertial reference frame, the
acceleration of an object is directly proportional to
the net force acting on it and inversely proportional
to its mass
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Force is the cause of change in motion, as measured by
the acceleration
Algebraically,
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a∝
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∑F
m
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→ ∑F = ma
With a proportionality constant of 1 and speeds much lower
than the speed of light
More About Newton’s Second
Law
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l  ∑F is the net force
l  This is the vector sum of all the forces acting on
the object
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Newton’s Second Law can be expressed in
terms of components:
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ΣFx = m ax
ΣFy = m ay
ΣFz = m az
Units of Force
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The SI unit of force is the newton (N)
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1 N = 1 kg·m / s2
Gravitational Force
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The gravitational force, Fg , is the force that
the earth exerts on an object
l  This force is directed toward the center of the
earth
l  From Newton’s Second Law
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l  F = mg
g
l  Its magnitude is called the weight of the
object
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Weight = Fg= mg
More About Weight
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Because it is dependent on g, the weight
varies with location
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g, and therefore the weight, is less at higher
altitudes
This can be extended to other planets, but the
value of g varies from planet to planet, so the
object’s weight will vary from planet to planet
Weight is not an inherent property of the
object
Gravitational Mass vs. Inertial
Mass
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In Newton’s Laws, the mass is the inertial mass and
measures the resistance to a change in the object’s
motion
In the gravitational force, the mass is determining
the gravitational attraction between the object and
the Earth
Experiments show that gravitational mass and
inertial mass have the same value