Transcript ISNS3371_012507_bw - The University of Texas at Dallas

ISNS 3371 - Phenomena of Nature
Angular Momentum
Momentum associated with rotational or orbital motion
angular momentum = mass x velocity x radius
ISNS 3371 - Phenomena of Nature
Torque and Conservation of Angular Momentum
Conservation of angular momentum - like conservation of momentum in the absence of a net torque (twisting force), the total angular
momentum of a system remains constant
Torque - twisting force
ISNS 3371 - Phenomena of Nature
The Moving Spool
Four forces: weight (mg), upward
normal force (N), tension in paper
(T), and friction force (N).
If spool not yet moving, net horizontal
force is zero or:
Tcos() = N
Only two of the forces produce a
torque about the center of the spool
(T and N). Equating the torques
gives:
r1T = r2 N
Dividing into previous equation gives
cos() = r1/ r2
This gives the critical angle which
determines which way the spool will
rotate
ISNS 3371 - Phenomena of Nature
Conservation of Angular Momentum
Conservation of angular momentum - like conservation of momentum in the absence of a net torque (twisting force), the total angular
momentum of a system remains constant.
Newton’s Third Law of Rotation Motion: For every torque that one
object exerts on a second object, there is an equal but oppositely
directed torque that the second object exerts on the first object.
ISNS 3371 - Phenomena of Nature
A spinning skater speeds up as she brings her arms in and slows down
as she spreads her arms because of conservation of angular momentum
ISNS 3371 - Phenomena of Nature
Angular Momentum
Momentum associated with rotational or
orbital motion:
angular mom = mass x velocity x radius.
The angular momentum vector is pointed
along the axis of rotation - right-hand rule: curl
the fingers of your right hand into a fist and
point your thumb up. If the direction of your
fingers is the direction of rotation, the angular
momentum vector is pointed along your thumb
Note: The angular momentum of a rigid body (a hoop, cylinder, etc…) is
the sum of the angular momentums of the particles composing the body
ISNS 3371 - Phenomena of Nature
Moment of Inertia
The property of a body that is a measure of its rotational inertia - resists a
change in angular (rotational) velocity (and thus angular momentum) analogous to mass - a measure of body’s translational inertia which resists
a change in translational velocity/momentum
- determined by mass and distribution of mass - how far the mass
is from center of rotation
Torque = moment of inertia X angular acceleration
This is analogous to F = ma
vt, at
 Angular acceleration measures
how fast angular velocity changes
r
 = vt/r
 is the angular
velocity
 = at/r
 is the angular
acceleration
so
 = r
ISNS 3371 - Phenomena of Nature
ISNS 3371 - Phenomena of Nature
Matter and Energy
ISNS 3371 - Phenomena of Nature
Matter
DEFINITION:
• Anything that occupies space and has mass
PROPERTIES OF MATTER:
• Mass - a measure of a body’s resistance to a change in its state of
motion - its inertia
• Density - mass per unit volume
• Dimensions - height, length, width
• Electric charge - positive/negative/neutral
• Heat content - everything above absolute 0 (-459.67º F) has heat no such quantity as cold - only absence of heat
• Resistance to flow of electric current - flow of charged particles electrons
• Pressure - exerted by moving molecules in all directions - resists
compression
ISNS 3371 - Phenomena of Nature
Energy
Definition of Energy:
• Anything that can change the condition of matter
• Ability to do work – the mover of substance (matter)
• Work is a force acting over a distance
• Force: The agent of change – push or pull on a body
Hence: Work is the change in the energy of a system
resulting from the application of a force acting over a
distance.
Work = force X distance
Units of Energy:
Joule = amount of work done when a force of 1 Newton is
applied over 1 meter
1 J = 1N - m = 1 kg m2/s2 1 Joule = 1/4184 Calorie, so
2500 Cal = 1 x 107 J (average daily requirement for a human)
ISNS 3371 - Phenomena of Nature
Energy Comparisons
Solar energy striking Earth’s surface per second = 2.5 x 1017 J.
Energy released by burning 1 liter of oil = solar energy striking square
100 m on a side in 1 second
ISNS 3371 - Phenomena of Nature
Fundamental Forces of Nature
Four Types of Forces:
•
Gravitational – holds the world together
•
Electromagnetic – attraction/repulsion of charged matter
•
Strong Nuclear – holds nucleus together
•
Weak Nuclear – involved in reactions between subatomic
particles
ISNS 3371 - Phenomena of Nature
Energy
Three basic categories:
Mechanical
Energy
{
Kinetic energy = energy of motion
KE = 1/2mv2
Potential energy = stored energy
gravitational, chemical, elastic,electrostatic, etc…
Radiative - energy carried by light
ISNS 3371 - Phenomena of Nature
Potential Energy
One form of potential energy is gravitational
potential energy - the energy which an object stores
due to its ability to fall
•It depends on:
– the object’s mass (m)
– the strength of gravity (g)
– the distance which it falls (h)
PE = mgh
Before the sun was formed - matter contained in
cloud diffuse gas cloud - most far from the center large gravitational energy. As cloud contracted
under its own gravity - gravitational energy
converted to thermal energy until hot enough to
ignite nuclear fusion
g
m
h
ISNS 3371 - Phenomena of Nature
Potential Energy
• energy is stored in matter itself
• this mass-energy is what would be released if an amount of
mass, m, were converted into energy
E = mc2
[ c = 3 x 108 m/s is the speed of light; m is in kg, then E is in joules]
The mass energy in a 1-kg rock is equal to as much energy as 7.5 billion
liters of oil = enough to run all the cars in the U.S. for a week
A 1-megaton hydrogen bomb converts only about 3 ounces of mass into
energy.
ISNS 3371 - Phenomena of Nature
Conservation of Energy
• Energy can be neither created nor destroyed.
• It merely changes it form or is exchanged between objects.
• This principle (or law) is fundamental to science.
• The total energy content of the Universe was determined in the
Big Bang and remains the same today.
ISNS 3371 - Phenomena of Nature
Types of Energy
Energy cannot be created or destroyed, only changed
–
Mechanical –
• Potential - stored energy
• Kinetic- energy of motion KE=1/2mv2
–
Electrical
–
Chemical
–
Elastic
–
Gravitational
–
Thermal
–
Radiant
–
Nuclear
ISNS 3371 - Phenomena of Nature
Conversion of Energy
Throwing a baseball
Nuclear energy (nuclear fusion on sun) - Radiative energy
(sunlight) - Chemical energy (photosynthesis) - Chemical energy in
pitcher’s body (from eating plants) - Mechanical kinetic energy
(motion of arm) - Mechanical kinetic energy (movement of the
baseball). Thus, ultimate source of KE in baseball is mass energy
stored in hydrogen of Sun - created in Big Bang.
Hydroelectric dam
Gravitational - mechanical - electrical
Nuclear reactor
Nuclear - thermal - mechanical - electrical
Car
Chemical - thermal - mechanical
ISNS 3371 - Phenomena of Nature
Power
Power:
Rate of change of energy
Power = work done/time interval = E/t
(remember:  means a change in a quantity)
Power:
1 watt = 1J/s
Thus for every second a 100 W light bulb is on, the electric
company charges for 100 J of energy.
The average daily power requirement for a human is about
the same as for a 100-W light bulb.
ISNS 3371 - Phenomena of Nature
Applications of Conservation of Energy
ISNS 3371 - Phenomena of Nature
Machines
Machines can be used to multiply force:
(force X distance)input = (force X distance)output
Decrease the distance and the force will increase.
Work/Energy is not changed!
ISNS 3371 - Phenomena of Nature
Levers
Fulcrum is in the center:
d1 = d 2
so
F1 = F2
Fulcrum is closer to one
end:
d1 > d 2
So
F2 > F1
Give me a long enough lever and a place to put the fulcrum and I can
move the world (Archimedes, 250 BC).
ISNS 3371 - Phenomena of Nature
Pulleys
ISNS 3371 - Phenomena of Nature
Pendulum solution (you are not
expected to know this)
Fnet  mgsin 
For small angles, sin = 
Fnet  mg  ma

mg  ml
This becomes the differential equation:
d 2 g
  0
2
dt
l

vt, at
r
 = vt/r
 is the angular
velocity
 = at/r
 is the angular
acceleration
so
 = r
With solution

g
  max cos t
l
For a complete oscillation:

g
P  2
l
so
l
P  2
g