Dimensionful Quantities

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Dimensionful Quantities

• Meters, seconds, light-years, kilograms, etc.
  are dimensions/units describing quantities.
• The dimensions of a quantity can be a
  combination of these, but must be the same on
  both sides of any equation.
• These qualify the numbers were talking about.
  Its not useful for me to say I weigh 155, but
  saying 155 lb is meaningful (and even that
  requires that you know Im on the surface of the
  Earth)

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Dimensional Analysis
Doesnt always work, but its typically a good
starting point look at the dimensions of the
numbers youre dealing with and the dimensions of
what you want to extract. Example If Im
traveling 75 miles/hour, and I drive for 2 hours,
how many miles did I go? ? Cancel out the units
/ build the units of miles (miles / hour) hours
? miles ? 75 2 miles 150 miles
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Vector Quantities
Now we add one more step of complexity, a
direction associated with the quantity, in
addition to is value. Example Marin is 15
miles from here (not a vector) vs. Marin is 15
miles NORTH from here (vector).
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Speed and Direction
Speed and direction are the two most basic ways
to quantify constant motion. Speed is obviously
just how fast youre going and direction just
defines the (instantaneous) path youre taking.
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Frames of Reference
To talk about distances, velocities etc., we
first need to choose a frame of reference. That
is, chose a point in space (space-time). Then
the velocities and positions we discuss are
relative to that point (or frame of
reference). Our typical frame of reference
corresponds to our eyes/bodies
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Position

• Two ways to look at this one
• As a scalar Im here, at this lat/lon
• As a vector Seattle is x kilometers North
  Northwest from here.
• Dimensions of position
• distance

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Velocity
This is the vector equivalent of speed, its just
speed in a specified direction. To measure
speed, we could use a pedometer and a stopwatch
(or just a speedometer) To measure velocity, we
need a compass also Dimensions of
velocity distance / time
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GPS
Whats a GPS? Anyone use a GPS? What information
does a GPS give you? Assuming it updates every
second, how does it give you a direction? How
does it compute your speed?
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Credit David A. Turner
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Average vs. Instantaneous Quantities
Two possible measures of speed/velocity/accelerati
on. Examples Instantaneous speed is what you
read off your speedometer. Average would be
the distance you traveled / time it took to get
there (or over shorter times, like with a gps)
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Acceleration
Acceleration is a measure of the change in
velocity, so its also a vector. Dimensions of
Acceleration distance / time / time
or distance / time2 (or velocity/time)
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Motion and Gravity
On the surface of the earth, everything feels the
same acceleration due to gravity
(approximately) However, other forces act on an
object (e.g. wind resistance) so when you drop a
bowling ball and a feather from the roof of a
tall building, only one is going to cripple
somebody!
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Mass versus Weight
Mass is an intrinsic quantity your mass is the
same on the Moon as it is on Earth. Its simply
how much matter you contain. Weight is actually
a force, and is a combination mass and the
acceleration due to gravity (and the weight of
the air above us, and the rotating motion of the
Earth, etc.)
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Freefall (falling w/o any resistance)
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Momentum
Momentum (inertia) is just a combination of the
velocity of an object and the mass of the object
thats moving, namely Momentum (often written
as p) mass x velocity ? It two people are
traveling in a car, they both have the same
velocity, but the heavier one will have more
momentum
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Angular Velocity
This describes the velocity of an object around
something (a point in space) and is equal to
distance x velocity (see below), (theres also a
factor for angle but we ignore that for this
course) Its a vector and follows the right
hand rule
v
r
Actually a circular path (this is a projection)
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Angular Momentum
This describes momentum around something (a point
in space) and is equal to distance x mass x
velocity (see below), also a factor for angle but
we ignore that here Its a vector and follows
the right hand rule
v
r
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Force
A force is required to change momentum/velocity.
The units of force are just momentum / time (the
rate of change of momentum over time). Force is
also a vector. Dimensions of force mass x
distance / time / time same units as momentum
divided by time.
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Net Forces
Most objects have more than one force acting on
them. In these cases, we can figure out all
the individual forces acting on an object, and
add them together and get a single net
force. Forces can cancel each other out
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Newton (1642-1727)
Gravity is universal similar to gravity pulling
objects towards Earth, the planets are effected
by gravity! Up to this point, it was assumed
that the physics of the heavens and Earth were
distinct. Lots of work on optics and
mathematics including building the first
reflecting telescope. And Newtons Laws of
Motion and Gravitation
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Newtons First Law
An object moves at constant velocity if there is
no net force acting on it.
Cruise control is one example of balanced forces.

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Newtons Second Law
Force Mass x Acceleration

• Examples
• 2 cars, same engine, different weight
• 2 cars, same weight, different engines

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Shot-put versus Baseball
Assuming you can apply the same force for the
same amount of time, which will go farther? (of
course, most people cant throw a shot-put like a
baseball) If a soccer ball weighs more than a
baseball, and you throw a baseball the same speed
as you can kick a soccer ball, which is stronger,
arm/shoulder or leg?
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Newtons Third Law
For any force, there is always an equal and
opposite reaction force.
This is why the bat breaks in baseball. Can you
think of an example of a force without a reaction
force?
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Newtons Laws

• An object moves at constant velocity if there is
  no net force acting on it.
• Force Mass x Acceleration
• For any force, there is always an equal and
  opposite reaction force.

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Conservation Laws
Based on Newtons Laws of Motion, we can show
that in a closed system, certain quantities
remain constant, or are conserved.
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Conservation of Momentum
The total momentum in any closed system will
remain constant. Closed means that there are no
external forces (in other words, all the forces
cancel with their equal and opposite forces) ?
Fsum 0 m1a1 m2a2 Momentum can be
transferred between objects in the system.
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Conservation of Angular Momentum
p constant m v r In this case, the mass
is constant and the velocity and radius change.
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Conservation of Angular Momentum
Explanation for Keplers Second Law
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Centripetal and Centrifugal Forces
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Energy

• Energy comes in several forms
• Potential (Chemical, Rest, Gravitational)
• Kinetic (motion)
• Radiative

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Units of Energy
The SI unit of energy is joules. The dimensions
of joules are mass x velocity x velocity
mass x distance x distance / time / time mass x
acceleration x distance force x distance
angular momentum / time torque
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Potential Energy
This is basically stored energy. It can be in
the form of gravitational potential (bucket of
water over a doorway), stored chemical energy
(firecrackers, bateries, gasoline) or rest energy
(WMD).
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Radiative Energy - Light

• Examples of light carrying energy (ALL light
  carries energy)
• Sunlight
• Microwave
• Lasers
• This is the focus of the next chapter.
• Radiometer Example

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Kinetic Energy
This is the energy of motion For those
interested, kinetic energy can be written as ½
mv2
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Temperature and Thermal Energy
Temperature is tough to describe Its really a
measure of average kinetic energy
Dots and arrows describe moving particles
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Temperature Scales
The zeropoints for Celsius are easy does
anyone know what 0 F corresponds to?
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Temperature and Density
Just as important as the temperature is the
density. Ill put my hand in a 500 F oven But I
sure as hell wont put my hand in 210 F water!
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Phase Transitions
Phase transitions occur when matter goes through
different states solid, liquid, gas,
plasma Energy also plays a part in phase
transition Energy is needed to melt ice or
vaporize water. But energy is released when
water vapor condenses. Phase transitions play an
important role in the early universe.
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Rest Energy or Mass Energy
This is the whole E mc2 thing This is the
source of energy in nuclear weapons. Stars also
tap this energy source through fusion.
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Conservation of Energy
The total energy in a closed system is
conserved. However, as weve just seen, energy
can be hidden (stored potential energy) in lots
of ways.