isns 3371 - phenomena of nature angular momentum momentum associated with rotational or orbital...
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ISNS 3371 - Phenomena of Nature
Angular Momentum
Momentum associated with rotational or orbital motionangular 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
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
The Moving Spool
ISNS 3371 - Phenomena of Nature
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.
Conservation of Angular Momentum
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 InertiaThe 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 = mavt, at
r
= vt/r is the angularvelocity
= at/r is the angular acceleration
so = r
Angular acceleration measures how fast angular velocity changes
ISNS 3371 - Phenomena of Nature
ISNS 3371 - Phenomena of Nature
Matter and Energy
ISNS 3371 - Phenomena of Nature
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
Matter
ISNS 3371 - Phenomena of NatureEnergy
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 meter1 J = 1N - m = 1 kg m2/s2 -
1 Joule = 1/4184 Calorie, so2500 Cal = 1 x 107 J (average daily requirement for a human)
ISNS 3371 - Phenomena of Nature
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
Energy Comparisons
ISNS 3371 - Phenomena 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
Fundamental Forces of Nature
ISNS 3371 - Phenomena of Nature
Energy
Three basic categories:
Kinetic energy = energy of motion
KE = 1/2mv2
Potential energy = stored energy
gravitational, chemical, elastic,electrostatic, etc…
Radiative - energy carried by light
{MechanicalEnergy
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
m
h
g
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 = mcE = mc22
[ 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 weekA 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
CarChemical - thermal - mechanical
ISNS 3371 - Phenomena of Nature
Power:Rate of change of energy
Power = work done/time interval = E/t
(remember: means a change in a quantity)
Power:1 watt = 1J/sThus 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.
Power
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 = d2
so
F1 = F2
Fulcrum is closer to one end:
d1 > d2
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
€
Fnet = −mgsinθ
For small angles, sin =
€
Fnet = −mgθ = ma
€
−mgθ = mαl
vt, at
r
= vt/r is the angularvelocity
= at/r is the angular acceleration
so = r
This becomes the differential equation:
€
d2θ
dt 2+g
lθ = 0
Pendulum solution (you are not expected to know this)
With solution
€
=max cosg
lt
For a complete oscillation:
€
g
lP = 2π so
€
P = 2πl
g