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Relative Velocity
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
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Different Measurements, example
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
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Different Measurements, another example 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 The combination of the
speed of the beltway and the walking
The difference is due to the relative velocity of their frames of reference
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Relative Velocity, generalized
Reference frame SA is stationary
Reference frame SB is moving to the right relative to SA at This also means that SA
moves at – relative to SB
Define time t = 0 as that time when the origins coincide
ABv
BAv
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Notation
The first subscript represents what is being observed
The second subscript represents who is doing the observing
Example The velocity of A as measured by observer B
ABv
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Relative Velocity, equations
The positions as seen from the two reference frames are related through the velocity
The derivative of the position equation will give the velocity equation
is the velocity of the particle P measured by observer A is the velocity of the particle P measured by observer B
These are called the Galilean transformation equations
PAu
PA PB BAt r r v
PA PB BA u u v
PBu
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Acceleration in Different Frames of Reference
The derivative of the velocity equation will give the acceleration equation
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
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The Laws of Motion
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Sir Isaac Newton
1642 – 1727 Formulated basic laws
of mechanics Discovered Law of
Universal Gravitation Invented calculus Many observations
dealing with light and optics
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Force
Forces are what cause any change in the velocity of an object Newton’s definition A force is that which causes an acceleration
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Classes of Forces
Contact forces involve physical contact between two objects Examples a, b, c
Field forces act through empty space No physical contact is required Examples d, e, f
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Fundamental Forces Gravitational force
Between objects Electromagnetic forces *
Between electric charges Nuclear force
Between subatomic particles Weak forces
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
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More About Forces
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
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Vector Nature of Forces
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
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Newton’s First Law
If an object does not interact with other objects, it is possible to identify a reference frame in which the object has zero acceleration This is also called the law of inertia It defines a special set of reference frames called
inertial frames We call this an inertial frame of reference
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Inertial Frames
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 We can consider the Earth to be such an inertial frame,
although it has a small centripetal acceleration associated with its motion
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Newton’s First Law – Alternative Statement
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 Newton’s First Law describes what happens in the absence
of a force Does not describe zero net force
Also tells us that when no force acts on an object, the acceleration of the object is zero
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Inertia and Mass
The tendency of an object to resist any attempt to change its velocity is called inertia
Mass is that property of an object that specifies how much resistance an object exhibits to changes in its velocity
Masses can be defined in terms of the accelerations produced by a given force acting on them:
The magnitude of the acceleration acting on an object is inversely proportional to its mass
1 2
2 1
m am a
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More About Mass
Mass is an inherent property of an object Mass is independent of the object’s
surroundings Mass is independent of the method used to
measure it Mass is a scalar quantity The SI unit of mass is kg
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Mass vs. Weight
Mass and weight are two different quantities Weight is equal to the magnitude of the
gravitational force exerted on the object Weight will vary with location
Example: wearth = 180 lb; wmoon ~ 30 lb
mearth = 2 kg; mmoon = 2 kg
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What is mass?
Gravitational force
€
F =Gmg1mg2r2
“Gravitational masses”, “characteristic constant” of the particles experiencing a gravitational force
€
F = ma
“Inertial mass”, another “characteristic constant” of the accelerated particle
Inertial force
€
GRAVITATIONAL MASS
INTERTIAL MASS=1 ?DOES
Answer: “Yes” macroscopically. “Maybe not” quantum mechanically!http://www.technologyreview.com/view/419367/new-quantum-theory-separates-gravitational-and-inertial-mass/
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Newton’s Second Law 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 Force is the cause of change in motion, as measured by
the acceleration Algebraically,
With a proportionality constant of 1 and speeds much lower than the speed of light
mm
F
a F a
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More About Newton’s Second Law
is the net force This is the vector sum of all the forces acting on
the object Newton’s Second Law can be expressed in
terms of components: Fx = m ax
Fy = m ay
Fz = m az
F
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Units of Force
The SI unit of force is the newton (N) 1 N = 1 kg·m / s2
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Gravitational Force
The gravitational force, , is the force that the earth exerts on an object
This force is directed toward the center of the earth
From Newton’s Second Law
Its magnitude is called the weight of the object Weight = Fg= mg
gF
g mF g
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More About Weight
Because it is dependent on g, the weight varies with location 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
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Gravitational Mass vs. Inertial Mass
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