Lesson 20 Angular Momentum

Kepler’s second law – an imaginary line from the sun to a planet sweeps out equal areas equal times   - applies to the vortexes of hurricanes, firestorms, and bathtubs, the center of this swirling motion is the law of conservation of angular momentum.

During the 16^th and 17^th century the Renaissance came to an end and Europe turned to Christian theology. This period was known as the Reformation, and afterward the Counter Reformation developed during the Thirty Years War. During this time, Galileo, Kepler, and Tycho Brahe all contributed to the study of the universe and physics. Kepler was a professional astrologist that caste horoscopes in order to make ends meet. Kepler almost discovered integral calculus over the dispute of the shape of a wine barrel, and eventually developed three laws that describe planetary motion.

Kepler’s second law in particular explains the shape of all galaxies as a factor of the conservation of momentum. Astronomy was the first physical science before medicine. In fact, early astronomers attempted to make a model of heaven and earth.

The Greeks made many models, including Aristotle, who in 300 B.C. placed the Earth at the center of the universe, as the sun and all other planets rotated around it in uniform circular motion. Astronomers continued to develop complex models with epicycles, however they were not correct. Copernicus later placed the planets around the sun, but kept the uniform circular motion of the bodies. This belief was scientifically accepted until Kepler found the three laws of planetary motion.

One law in particular (Kepler’s second law) states that a planets orbit sweeps out equal area in equal time.

The area of the parallelogram formed by the two vectors r and (delta) r is the cross product.

(delta) A = ½ r x (delta) r because half the area of the parallelogram would be equivalent to the triangular area swept out by the planet’s orbit.

(delta)A/(delta)t = ½ r x (delta)r/(delta)t

dA/dt = ½ r x v is equivalent to the rate of change of the area vector, which is constant in all parts of a planet’s orbit. This means that as the distance r increases, velocity decreases and as the distance r decreases, velocity increases.

When momentum is angular there is a twist known as torque. Even if force is applied to a body, the torque may be zero. If the twist is zero than vector r x F = 0

Gravity and fluid pressure are the main influence on vortexes.

F = 0, therefore integral was constant p

dL/dt = r x F

d( r x v)/dt = dr/dt xv + rx dv/dt

= r x dv/dt

= r x a

F = ma

d(mr x v)/dt = r x ma

= r x F = T or torque

L = mr x v

r x F = 0 so mr x v is constant as angular momentum constant L

A planet in orbit has no torque so it conserves angular momentum. That’s why Kepler’s second law is true. According to the law of inertia, a body moving in a straight line will continue at constant speed unless a force is applied to it. In a similar way, a spinning body rotates in the same direction unless a twist or torque is applied. This is the essence of the law of the conservation of momentum.

Angular momentum L is a vector following the right hand rule

L = mr x v

L = mrv when v and r are perpendicular

If orbit is smaller speed must increase to maintain L as a constant.

Bathtubs, tornadoes, and hurricanes all exhibit angular momentum.


Water lacks enough viscosity to apply torque. In tornadoes calm air towards the center moves upward forcing circulating air upward to replace it. Jupiter’s Red Spot is an enormous hurricane that has lasted for at least three hundred years since it was first observed, due to the principles of inertia. Furthermore, the Solar System, the planets, and the galaxies exhibit flat disc shaped planes with globular masses at the center due to the conservation of angular momentum.

Below is a video that explains the concept of angular momentum in depth:

Lesson 19 Torques and Gyroscopes

When a torque acts on a spinning object the angular momentum changes. The rate of change of angular momentum is equal to the applied torque. Under the influence of a torque, a spinning object precesses.

All conservation laws come from F = m a

Newton's Mechanics based on three laws, however everything we do is based off of F = m a

Newton's third law implies conservation, thus a torque on one body implies a reverse torque on another. Angular momentum flows from one body to another and angular momentum is conserved.

The advancement of cultures over the course of history can be traced by advancements in transportation. Transportation dramatically changed with the creation of the wheel.

The force of gravity naturally tips the wheel down. However,  this is different with a spinning wheel.

When there is angular momentum there is a twisting force that changes the angular momentum. When angular momentum is conserved r x F (the radius vector and the force vector) is conserved.

A spinning wheel already has angular momentum, and torque can change that.

T = dl/dt

Below is a video that explains the concept of torque:


Gravity plays a strange trick on a spinning wheel. The force of gravity is downward, however the force of torque is sideways. This is why a spinning wheel will precess when stationed on an axis. When gravity tries to make a non-spinning wheel fall, the spokes push the top of the wheel outward rather than downward, which causes the weel to fall. This creates a component of velocity in the outward direction. This same thing happens at the bottom of the wheel but the force of gravity pulls the wheel inward.

This same idea also occurs when the wheel spins, however, when the outward velocity builds up it is at the side of the wheel. The force is outward on one side and inward on the other -- each spoke exerts its own push at the top -- causing a jog at the side. The net result is a downward but sideways motion when the wheel spins (precessing).


A top and bicycle wheel are a lot alike. A top acts as if all its mass is concentrated at one point, known as the center of mass. When the top is tilted there is a torque created. When this happens to a spinning top the torque nudges the anguar momentumof the top around in a little circle. Heat is produced at the tip of the top where energy is transfered into a random motion of molecules.

If a gyroscope is balanced at its own center of mass, the force of gravity will not make it precess. Even if it supports swivel in all directions the axis will remain stable and keep the gyroscope pointed in the same fixed direction.


Gyroscopes are used in planes in order to fly over magnetic poles using computer aide. Gyroscopes are improved by removing torque.

:C or capital omega is the rate of precession

U = wr

T = :C L

:C = Rg/wr^2

:C is just torque divided by angular momentum.

An object will precess least if wheel is large, spinning fast, and carefully balanced.

The best gyroscope on earth is the earth itself. The plane of the equator where day and night are equal length intersects the earth's orbit at the equinoxes. These points are used to denote seasonal changes and slowly dift to the west each year in the ever expanding universe.

Lesson 18 Waves

Oscillation can travel through mediums such as air or water without carrying water along with them. These mechanical disturbances are called waves. Waves are one of the most common natural phenomena.


Sound travels very fast but it is not infinite. Sound is a wave or disturbance that travels at a definite speed.

The "Big Bang" is an example of a wave where a disturbance in one place led to reactions in another.


When any stable mechnical system is disturbed nature's response is simple harmonic motion, where the disturbance in one place is passed on to the next as a mechanical wave.

In a crystaline solid, linkage is weak where disturbances travel slowly, and linkage is strong where disturbances travel rapidly. Any waves that propogate through a medium are mechanical waves. Impulses pass through a crystal from atom to atom because each atom is bound together to an equilibrium position by electrical forces. These atoms act mechanically like masses connected by springs.

Musical instruments send waves through the air with an amplitude the size of the disturbance which is preserved as the waves moves along. The time for each complete cycle is known as a period. The inverse of this is the frequency on which the tone of an instrument depends. The definite distance from one compression to the next is known as the wavelength.

wavelength = T v (speed of wave)

frequency x wavelength = v where speed is always the same.

In the case of water waves, speed decreases as they approach the shore. Long waves travel faster than shorter waves, however all waves obey the same principles.


Harmonic oscillations respond and spring back.

For masses on a spring:

v = a sqrt(k/m)

Gravity and the length of the water wave determine speed.

v = sqrt ( g (wavelength) / 2pi)

Masses on a spring produce waves in the direction of connection known as longitudal waves.

Waves that travel throughout a solid are known as transverse waves, however water waves are classified in neither of these categories.

Each water molecule moves on the surface in little circles where each circle is slightly offset from the next.

v = sqrt (g h) in shallow water

Waves of sound vibrate air into motion, expanding and compressing density with each vibration. This is why sound carries the same frequency as the source driven by the force that produces change in pressure as density increases and decreases. Below is a video that physically displays sound waves:



v = sqrt (pressure/density)

Newton attempted to find the speed at which sound traveled along the corridors at Trinity University and determined that the sound waves traveled at a speed of 979 ft/sec. However, Newton was incorrect and William Derham discovered that the speed of sound was in fact 1142 ft/sec. This discrepency is due to the fact that air heats up as it is compressed and travels back faster. This was not discovered until a century later.

Below is a video that explores the nature of waves in depth:

Lesson 17 Resonance

When a force is applied repeatedly at the natural frequency of any system, large amplitude oscillations result, this is the phenomenon of resonance.

On July 1^st, 1940, a delegation of citizens met in Washington states to engineer the Tacoma Narrows Bridge. This bridge was the longest suspension bridge in the world, but was destroyed due to the force of resonance. The word resonance itself is the Latin noun for “echo.” Below is footage of the Tacoma Narrows Bridge:

With music, the pleasant sounds are strictly due to the resonance of the instrument. All instruments resonate equally no matter what note is played. On the violin or cello, the strings vibrate hundreds to thousands of cycles per second and cause the whole instrument to vibrate at the same frequency. Every vibration gives the instrument’s “sound box” a push so that the “sound box” vibrates resonantly. The vibrations excite resonant vibrations of the air inside the sound box.

A tuning fork is a common tool used to tune the piano. How does this device function? The tuning fork sets air molecules into harmonic motion. Air works as an elastic medium that is a good conductor of traveling oscillations waves. The tuning fork pushes air molecules into waves to the natural frequency of the tuning fork. When the pitch of the piano is equal to the pitch of the tuning fork, the waves from the piano can disturb its Naturally oscillating systems. Below is a video of a tuning fork being used to tune a piano:

Resonance obeys the differential equations of a mass on a spring:

F = -k x

m a = -k x

m (d^2x/dt^2) = -k/m x                              x = c sin (omega(0) t)

dx/dt = (omega(0)) c cos(omega(0) t)

d^2x/dt^2 = (-omega(0)^2) c sin (omega(0) t)

(-omega(0)^2) c sin (omega(0) t) = -k/m (c sin (omega(0) t))

omega(0)^2 = k/m

Due to the repetitive force applied while “pumping” on a swing, this motion is considered to be a form of resonance.

F = -k x                

F(0)sin (omega t) where the force applied at a different frequency is proportional to F

F =  -k x + F(0)sin (omega (t))

m a = -k x + m a(0) sin (omega(t))

d^2x/dt^2 = -kx/m + m/m a(0) sin (omega (t)) = -(omega(0)^2 x + a(0) sin (omega (t))

x = c sin (omega(0) t) + A sin (omega (t))

d^2x/dt^2 = -(omega(0)^2 c sin (omega(0) (t))) – (omega^2 A sin (omega (t)))

-(omega(0)^2 c sin (omega(0) (t))) – (omega^2 A sin (omega (t))) = -(omega(0)^2 c sin (omega(0) (t))) – (omega^2 A sin (omega (t))) + a(0) sin (omega (t))

A sin (omega (t)) = a(0)/ (omega(0)^2 – (omega)^2) sin (omega (t))

The size of the oscillations depends on the size of the force a(0) and how close its frequency (omega) is to the natural frequency omega(0).

The amplitude of forces oscillations is sensitive to the frequency.

A = a(0)/ (omega(0)^2 – (omega)^2)

If the frequency of an object is close to its natural frequency, spectacular things can happen. On of the most famous is the shattering of a wine glass due to the frequency of a persons voice. Here is a video that demonstrates the shattering of a glass:

It is important to note that glass is a viscous fluid. If glass was not viscous, there would be no such thing as a glass window. Forces can be sent out at many frequencies. One of the most detrimental forces are earthquakes. The force of earthquakes resonate at a very low frequency that causes buildings to shake and deteriorate as shown below:


Aoelian harps function at certain wind speeds, which cause vorteces to peel off at the relative frequency of the wire. The "sea organ" below functions similarily:


Most systems containing vorteces are all examples of resonance, an important aspect of physics.

Lesson 16 Harmonic Motion

Newton’s second law F = m a combined with a restoring force leads to a motion that repeats itself over a period of time. This is known as harmonic motion. Galileo studied a lamp in a church and timed the lamp’s swing. Galileo observed that each swing took the same amount of time even as the motion died down. This observation led to the development of the first accurate clock. In other words, pendulums can be used to keep time.

Time must be “kept” in music which is also an example of harmonic motion. Musical instruments share a special property with bobbing weights. Both the weights and instruments create vibrations at a certain frequency which produces a certain pitch that doesn’t change as the motion decreases. There are many factors that determine the tone of a musical instrument including: length of string, size, shape, and the technique with which the instrument is played. In the physics behind music, one factor never varies. Once the tone has been produced the pitch is the same even though the vibrations diminish. The pitch remains constant because of force, the F in the equation F = m a.




In a certain position all forces are balanced, however, when a spring is stretched it attempts to pull the mass back to its original position. The farther the mass moves the greater the force to pull the mass back to its original position. Conversely, when a spring is compressed it tries to push the mass back, which ever direction the mass moves, a force emerges to counter the displacement. The combination of this force and the mass’s inertia will be the key to keeping time.

At each point in its motion, the net force is proportional to the distance x of the mass from its equilibrium position and in the opposite direction.

The equation is F = -k x where the value of k depends on how stiff the spring is.

In earlier periods of history, there was a constant need for objects that keep accurate time. Cycles in nature, lunar eclipses, solar eclipses, and solar cycles were all used to tell time. However, with the discovery of harmonic motion, clocks were developed to keep accurate time because all clocks oscillate with precision following the most inexorable law in physics F = m a 

F = m a
F = m d^2x/dt^2
F = -k x
d^2x/dt^2 = -k/m (x)

This equation describes the simple harmonic motion of not only a mass on a spring but any system that when disturbed, returns to equilibrium with a force proportional to the distance.

x = A sin(omega(t))

dx/dt = A omega cos(omega(t))

d^2x/dt^2 = -A (omega)^2 sin(omega(t))

d^2x/dt^2 = -w^2 x

d^2x/dt^2 = -k/m (x)

(omega)^2 == k/m

(omega) = sqrt (k/m)

Frequency = omega = 2pif where omega (radians per second) is equal to 2pi (radians cycle) multiplied by the frequency (cycles per second).

Principles of uniform circular motion and harmonic motion are closely related as viewed in the video below:

Potential energy of harmonic motion can be modeled by a curved shaped bowl.

Potential energy = du/dx = -F

integral(du/dx) = kx

U = k(½)x^2 while K = (½) m v^2

As a ball rolls in the curved shaped bowl there is  a continual interchange between potential and kinetic energy.




U = k(½)x^2                                K = (½) m v^2

x = A sin(omega(t))                     v = omega A cos(omega(t))

U =  k(½) A^2 sin^2(omega(t))

K = (½) m omega^2 A^2 cos^2(omega(t))

omega^2 = k/m

m(omega)^2 = k

E = U + K

U+K = ½ k A^2

E =  ½ k A^2 where the total energy is constant although potential and kinetic energy change.
It takes the same time for a pendulum to complete a swing even if motion dies down. All pendulums of the same length oscillate at the same rate regardless of their mass. In the case of a falling body, if all bodies fall at the same rate, regardless of their mass, then all pendulums should oscillate at the same rate regardless of mass. All bodies fall to earth with the same constant acceleration. The force of gravity, which depends on the mass of the pendulum, will cancel the mass in the equation F = m a. In general, if an object is disturbed it exhibits simple harmonic motion. Below is a clip from the Mechanical Universe that explains harmonic motion:

Lesson 15 Conservation of Momentum

The momentum of an object is m x v; when no external forces act on a system the total momentum is constant, this is known as the conservation of momentum. The total quantity of motion in the universe is fixed. In other words, if a body is not interferred with it will move at a constant speed in a straight line (Inertia). The body does not come to rest but transfers motions to other bodies; the total energy is conserved. The change in motion is proportional to the force impressed, and is made in the same direction of the straight line in which the force is impressed.

Motion = Momentum

p = m v

Momentum = Mass x Velocity

All of Newton's laws can be represented by differential equations:

F = dp/dt (second law)

A single particle of some mass has p = m v

F = d   p
       dt

No force 0 = d   p
                     dt
p is constant and m v is constant as the motion of the object is constant

Billiards is a game where the conservation of momentum heavily applies. For instance, if a ball is traveling in a straight line (second law, inertia) it has momentum. If two billiard balls come in contact with each other, each ball applies a momentary force causing momentum to change. According to Newton's third law, this change is equal and opposite.

F1 = -F2
dp1/dt = -dp2/dt
d/dt(p1 + p2) = 0 because the total momentum of two balls taken together does not change and is constant.

This rule applies to any number of balls and also to the atoms of which the balls are made. Momentum is always conserved. Each ball is composed of atoms and smaller parts supplying equal and opposite forces to each other. Each ball behaves as if it were a single body with all its mass concentrated at a single point. This is known as the center of mass, where focus must be drawn when calculating velocity and acceleration of a compound body when no net outside force acts on a compund system. No matter what happens to its individual parts the center of mass continues to move at a constant speed in a straight line. Recoil, as when shooting a shotgun, is a prime example of conservation of momentum.



Below is a video that shows how the center of mass moves at a constant rate with the conditions stated above:


The conservation of energy has many forms, as enrgy is conserved strictly and absolutely no matter the masses of the bodies, or type of energy.

The sum of the momentum of all bodies is a conserved quantity that is always the same even with two or more bodies. When there is no outside force, the center of mass moves at a constant velocity. For instance, a planet pulls a moon just as the moon pulls the planet with equal and opposite force.


F1 + F2 = 0, when the derivative of something is zero, the original value was constant.

p = m v

v = p/m

Kinetic energy: k = 1/2 m v ^2
                          k = 1/2 m (p/m x p/m)
                          k = p^2/2m

One ball strikes another, this reveals that at rest, there are only two possible outcomes:

1.) p0 = p1 + p2

This is a vector equation that form a triangle. In the collision of billiard balls, only kinetic energy is observed (ignore heat from collison), and kinetic energy is also conserved.

k0 = k1 + k2           k = p^2/2m
p(0)^2/2m = p(1)^2/2m + p(2)^2/2m

p(0)^2 = p(1)^2 + p(2)^2 (The pythagorean theorem)

p(1) is perpendicular to p(2), so when one ball strikes another, the balls come off at right angles.

p(0) = p(1) + p(2)             p(0)^2 = p(1)^2 + p(2)^2

2.)  p(0) = p(2)                  p(0)^2 = p(2)^2

The ball transfers its momentum to the other ball and stops.




Physicists use collisions as the sole method for studying the subatomic world.

Descartes chose to look at nature in terms of mathematics and pictured philosophy as a tree. Where the base of the tree is "physics" with all other sciences branching from it.

Below is a video that explains the conservation of momentum in mathematical terms:

Lesson 14 Potential Energy

In 1711 Ruđer Bošković, a diplomat, architect, scientist, and writer, proposed the idea that all matter is made up of point masses, and when masses are far apart, force acts between the two masses. Bošković was in fact correct. For example if a force is graphed against position, the zeroes of the graph are points of no force or equilibrium. However, these points of equilibrium are both stable and unstable. A positive force is repulsive while a negative force is attractive. Therefore, a point with a negative force on the left and a positive force on the right is at unstable equilibrium while a point with a positive force on the left and a negitive force on the right is at stable equilibrium. Here is an image of Bošković's work below:

 From these findings, it was believed that atoms are bound into stable webs of matter, however, this was not a scientific idea because it could not be readily tested. In the following centuries, John Dalton discovered atoms while Michael Faraday studied fields of force. From these advances it was found that every piece of matter in a solid is in stable equilibrium, unlike the webs of stability predicted by Bošković.

The total amount of energy never decreases because energy is strictly and absolutely conserved. Potential energy depends on position, such as a tank of gasoline, where the positions of the atoms in the molecules of the gasoline determine the potential energy becasue an atoms potential energy depends on its position in the matter it's bound to. Potential energy can be both positive and negative without changing effect where F = -du/dx. The higher above earth an object is, the more gravitational potential energy the object has because potential energy changes with the objects position. Therefore, the closer to the ground an object is, the less potential energy it has.

Gasoline:

Kinetic energy is related to speed, where the faster an object travels the more kinetic energy it has while the slower an object travels the less kinetic energy it has.

Any body can trade potential energy for kinetic energy by trading position for speed.

Here is a video that displays the conservation of energy as potential energy tranfers into kinetic energy:



Molecules in oil have great potential enrgy, as the atoms are bound together by the electrical force between atoms. When the oil atoms are ignited, the molecules rearrange themselves into other forms of matter with less potential energy.

Here is a video that displays the enormous amount of potential energy of oil molecules:



On the other hand, a fire hose has initial potential energy as the wtaer enters the hose, kinetic as the water leaves the hose, and potential energy as gravity influences the path of the water. Work can be described as a change in energy:

Take the lifting of a vessel from one height to another:
          
               Rf
W = integral (Fdr)                            F = G m Me
               RE                                                 r^2
              
                Rf
     = integral (G m Me  dr)
                RE        r^2
                        
                           Rf
     = GmMe integral (dr/r^2)
                           RE

                          Rf
      = -GmMe/r |
                          RE

       =  -GmMe/Rf - (-GmMe/RE)

    u = -GmMe/r

W = uf - uE of the change in potential energy

If the ball is lifted to infinity it has a potential energy of 0, and also on the surface or at ground level.

(-GmMe/RE) + 1/2 (mv^2) = 0  because kinetic energy must balance out potential energy.

Therefore, in the case of a rocket leaving the earth's surface u + k = 0 to escape from the earth and reach infinity where:

V = sqrt ( 2 G Me/Re ) and the rocket will escape is traveling at 11 km/sec

In the case of a firefighter:

Potential Energy = mgh                                      m = 90kg of firefighter
                                                                             J = 10 m/s^2
                                                                             The height of the burning building is 30m
                            = 27000J
420 = 1 calorie of heat

About 6400 calories

However 1 food Calorie is equivalent to 1000 heat calories, therefore it would take to firefighter about 6.4 Calories of food storage.

Work = Force x Height

The human body is actually very inneficient because it uses most of its stored energy to mantain basic bodily functions. In the terms of physics, potential energy is relative to the ground, therefore if one climbs a ladder and returns to the ground, no net work has been accomplished.

Here is a video that describes the distinct difference between kinetic and potential energy:





                 

Lesson 13 Conservation of Energy

It is impossible not to conserve energy because energy is always conserved. This is a fundamental principle of physics. There are three parts to the Law of Conservation of Energy including: energy, momentum, and angular momentum. What is energy? Energy is observed in many shapes and forms on earth and the universe. However, if energy is always conserved how do objects get started?

W = Fh
Weight = Force x Height

If the object is near earth then W = mgh due to the force of gravity.

Therefore U = mgh or potential energy is equivalent to the "work put into it," where potential energy depends on vertical distance. Take an inclined plane for example:

The potential energy of the object depends on the height of the inclined plane. In a system with two adjacent inclined planes without any other external forces exerted on the object, the object would travel down the inclined plane where potential energy would change into kinetic energy, and then change into potential energy as it reaches the same initial height on the second inclined plane.

dW = Fdx

W = integral (Fdx)

If work is done against a constant opposing force, as in lifting a block from one height to another then:
                    h1
W = F x integral (dx)
                    h0
              h1
     = Fx |
              h0

   = Fh1 - Fh0

U = Fh

W = U1 - U2

The work is the difference between the potential energy at the two heights or the change in potential energy.

If work is done with no opposing force, then the force is still work integrated over disance where:
              x1
W = integral (Fdx)
              x0

however the result of the work is to accelerate the block, therefore it gains speed. If the interval is considered in terms of speed:

W = integral (madx)  F = ma

W = integral (m (dv/dt) dx)

     = integral (m (dx/dt) dv)

     = integral (mvdv)
                   v1
W = m (integral (v) dv)
                   v0
                          v1
     = m 1/2 v^2 |
                          v0
                   2             2
W = 1/2 m v  - 1/2 m v
                  1              0
Therefore work is the change in the quantity k = 1/2 m v^2 or kinetic energy, the energy of motion.

Potential energy changes constantly while kinetic energy is constantly in flux. When the sum of both is considered, the totality of energy is a constant:

E = U + K

energy = potential energy + kinetic energy

            = constant

The transfer of potential and kinetic energy can be observed in the following video:



 James Joule is credited with the discovery of the conservation of energy by studying how much mechanical energy turns into heat as a large weight is lifted to a certain height. This has a precise potential energy:

U = mgh

Joule arranged to have falling weights turn paddle wheels in an insulated container of water. Joules then measured the water's temperature, given the loss of potential energy always turns into precisely the same amount of heat because loss is also gain in energy. Vibrations of atoms have a number of effects by jostling air that creates soundwaves, and generates heat. Heat spreads from molecule to molecule dispersing energy but not destoying it. Energy is neither created or destroyed. Heat dispersion is merely spread into vibrations that increases potential and kinetic energy of atoms by exactly the amount that has been lost. Activity can be measured in the form of the same basic unit known as a Newton meter or 1J.

4.2 J = 1 cal, the standard measure of heat.

Energy flows throughout the universe ever changing but constant, and is not lost but transformed into heat. As energy changes forms, it becomes harder and harder to retrieve at the atomic level.

The following video explains the conservation of energy further:

Lesson 12 The Millikan Experiment

All the forces and quantities of matter can be divided into their individual parts. For example the photon (light), the phonon (sound), the proton (matter), and electricity (electron) are all example of individual units. J.J. Thompson was born in 1856 and lived until 1940. Thompsonn was not only an English pysicist, but also a Nobel laureate who created the mass spectrometer and discovered the electron and the existence of isotopes. However, Thompson is heavily associated with his use of the Cathode Ray Tube experiment to discover the electron. A cathode ray tube is a vacuum tube with electrodes on each side. An electrical current is sent through the vacuum tube creating a "ray" that can be deflected by both a magnet and and electrical field. Thompson realized that the experiment could not be conducted because the tube contained traces of gas left by the glass-blower. Thompson then heated the tubes, which allowed the traces of gas to leave the tube. Once the vacuum was achieved, Thompson's used a magnet/electrical field to study the deflection of the "ray." From this, Thompson concluded that the "ray" was composed of a negatively charged particle, the electron, and the experiment was successful.  Below is an animation of a cathode ray tube:


After the electron was discovered, Millikan would be the physicist to measure the charge of an electron using oil, an iron pot, and electrical power. However, during this time period there was a strong influence from Europe, regarding advances in the sciences. These advances fueled Millikan's ambition and included: Nobel Prize winner and Univeristy of Chicago professor Albert Michelson's creation of the instrument to measure distances by the interference of light waves; William Ramsay's discovery of the elements helium, xenon, krypton, and neon; William Rontgen's discovery of x-rays; Guglielmo Marconi's adaption of the wireless telegraph; Marie Curie nobel prizes form her work in physics and chemistry; Albert Einstein's Theory of Relativity; work by Max Planck and Johannes Stark; and J.J Thompson and A.J. Wilson's discovery of the effects of x-ray radiation and the cloud chamber method for measuring charge. With all these advances at the frontier of physics, Millikan was determined to use his skills with precise measurments in order to push the frontier slightly farther.

Therefore, Millikan performed the famous Oil-Drop experiment. Millikan first ionized a gaseous cloud, and utilized both x-rays and radium. Millikan then used a powerful electrical field to balance the force of gravity and suspend the cloud of water without motion. However, this presented a problem because the water cloud would evaporate. Therefore, Millikan used oil droplets that would not disappear in order to detect the effect of an electron. Millikan theorized that if an electrical field is applied to an electrically charged oil falling through the air, then the resulting charge of the electron could be determined using F=ma:

ma = summation (F)

F(gravity) = mg and viscosity

F(viscosity) = 6(pi)Rnv
where R = the radius of the sphere, n = the viscosity of the air, and v = velocity.

F(viscosity) is a constant speed, however the velocity grows until viscosity balances gravity.

v = mg/ 6(pi)Rn

Below is a video showing the principle of viscosity:


Millikan then needed to measure the speed in order to find out how big each drop of oil was because they were to small to see with the human eye:

D(density) = m/(4/3)(pi)R^3                                      mg/6(pi)Rn = v
R^2D = m/(4/3)(pi)R                                                 m/(pi)R = 6nv/g
(4/3)R^2D = m/(pi)R

(4/3)R^2D = 6nv/g
R = sqrt (6nv/(4/3)gp)

The electric field creates an electric force so F(electricity) = electric fields strength times the strength on the oil drop. F(e) = qe because the force of electricty will be an integer multiple of the fundamental unit of electricity.. The electric field will drive the drop up until a constant speed is reached.

qE - mg = 6(pi)Rnv

v = qe - mg/6(pi)Rn

v (electric field on) = qe - mg/6(pi)Rn                   v (electric field off) = mg/6(pi)Rn

v(on) + v(off) = qe/6(pi)Rn

qe = (v(on) +v(off)) 6(pi)Rn

q = (v(on) +v(off)) 6(pi)Rn/e

Millikan took meticulous steps in order to ensure that his experiment yielded the correct result. Millikan minimized the turbulence of the droplets between the two plates by housing the plates in iron pots. The air was filtered through glass wool and an atomizer was used to spray the finest mist of oil droplets into the chamber. Light was filtered out by a solution of copper sulfate that coated a one meter tube of wate to eliminate the heat of water. Below is a video displaying actual footage of Millikan's experiment repeated:



Millikan found that the charge of an electron was 4.77 x 10^-10 or 1.602 x 10^-19 coulombs, a two percent difference from the prior experiment performed by Thompson. In 1923, Millikan was the first native born American to get the Nobel Peace Prize in physics.

Lesson 11 Gravity, Electricity, and Magnetism

Three values govern the nature of the world. These values include: the force of gravity, the speed of light, and the electric charge of an electron. The Gravitational constant (G) is equivalent to 6.7 x 10^-8 dyne cm^2/g^2, a value so small that scientists do not have the appropriate technology to measure the force exactly. The speed of light (c) is equivalent to 3 x 10^10 cm/sec, and serves as a tool to measure the vastness of space. In other words, distance in space is measured by light-years or the distance light travels in one year. The electric charge of an electron (e) is equivalent to 4.7 x 10^-10 esu. These values were not discovered outright by a single physicist or scientist. Instead, numerous physicists contributed to these findings:

In 1675, Ole Romer observed the delay of the eclipse of Jupiter's satelites in order to calculate the velocity of light.

In 1849, Hippolyte Fizeau used rotating wheels to measure the speed of light through air and water.

In 1850, Jean Foucault named and improved the gyroscope and discovered the speed of light with a series of rotating mirrors.

In 1926, Albert Michelson measured the time it took light to travel from two large peaks (Wilson/San Antonio) in Los Angeles. Michelson found that it took .0001 sec for light to travel from one peak to the other. These peaks were roughly 30 km apart. Therefore speed is equal to distance/time, and the speed of light is about 3 x 10 ^8 m/s. Below is a video displaying how the speed of light can be measured using a chocolate bar and a microwave:

Another distinct force is magnestism:

F(m) = k(m) p1p2r^
                         r^2

Magnetic poles, represented by p1 and p2, are always in pairs that are equal and opposite. This equation is strikingly similar to the equation of the force of electricity. In 1820, Hans Christian Oersted discovered this relationship between electricty and magnetism known as electromagnetism. Oersted found that a compass needle would deflect from magnetic north when an electric current from a battery was switched on and off. Oersted first believed that magnetic effects were produce from all sides of the wire, but after future investigation, he discovered that an electric current produces a circular magnetic field as it flows through a wire. Below is an example of his experiment:



Gravity:
G = 6.7 x 10^-11 Nm^2/kg^2

Electricity:
K(e) = 9 x 10^9 Nm^2/e^2 (coulombs)

Magnetism:
K(m) = 1 x 10^-7 Ns^2/e^2 (amperes or coulombs per second)

Due to the relationship of electromagnestism, James Clark Maxwell discovered that magnestism and electricty were not independent:

K(e)/K(m) = 9 x 10^9 Nm^2/e^2
                     1 x 10^-7 Ns^2/e^2

                   = 9 x 10^16 m^2/s^2
This is a squared speed due to squared distance over a squared time, therefore the square root must be taken in order to obtain the speed:

                     =3 x 10^8 m/s  or the speed of light

Samuel Morse, the creator of the telegraph, commented on the complexity of the construction of a telegraph wire across the Atlantic Ocean. Such complex tasks compare to the equally complex task of discovering the relationships between all motion, matter, and force, the study of physics.

Lesson 10 Fundamental Forces

The variety of phenomena in the universe can be described by four forces: the strong nuclear force, the weak nuclear force, the electrical force, and the force of gravity. These are the four fundamental forces of nature, and every piece of matter is subject to these forces.

The first two fundamental forces of stong nuclear force and weak nuclear force are located in the nucleous of an atom.

The strong nuclear force is represented by neutrons in the nucleous that overcome the natural repulsion of protons, and bring the protons together. The power released in nuclear reactions is also due to the strong nuclear force. Reactions such as this are present on the sun.

The weak nuclear force has subtle effects in the nucleous and can be represented by the death of stars and the decay of radioactive isotopes. Below is a video that explains the two forces in great detail:



Gravity accelerates mass and reaches out over the bounds of the entire universe. Gravity is explained in terms of force which is measured in newtons.

In 1798, Henry Cavendish weighed the earth and was able to measure the constant in Newton's Law of Universal Gravitation (the force between any two objects):

F = ma                                                  F = mg

F = kg (m/s^2)                                     F = mm(e)
                                                                     r(e)^2

   = N                                                    Gm(e) = gr(e)^2         9.800 m/s^2

                                                             Gm(e) = (9.800 m/s^2) r(e)^2

                                                              r = (6.378 x 10^6 m)^2

                                                              Gm(e) = 3.986 x 10^14 Nm^2/kg

After finding Gm(e), Cavendish was able to weigh the earth using a device depicted below:

In this device, the gravitational force between the two masses causes the bar to twist. By timing the beam's oscillation, Cavendish measured the inertia of the beam with the two balls. The attraction to the other balls could be measured by deflection of the beam and Cavendish found G to be about 6.67 x 10^-11. Cavendish then took Gm(e) and divided it by G to find that the mass of the earth was 5.976 x 10^24 kg.

In regards to the force of electricity, everything is at equilibrium. For example, protons have a positive electrical charge, neutrons have a neutral electrical charge, and electrons have a negative electrical charge. However, all elements are stable with an equal amount of protons and electrons to produce an overall neutral charge.

The force of electricity:

F(e) = k(e) q1q2 r^
                     r^2
where k is the electrical constant, q1 and q2 are the charges, and r^2 is the square of the distance between two objects.

Spring action, friction and viscosity are all electrical forces. A metal spring exhibits electrical force because positively charged metal ions are glued by negatively charged electrons, and as the positive metal ions pull the spring apart, the negatively charged electrons pull the spring back together. Friction is the binding of positive and negative forces as they interact, and viscosity exhibits the interaction of negative and positive forces.

All of the forces mentioned above can be observed in ion and proton accelerators, which are commonly used to study the origns of stars. Friction, gravity, strong nuclear force, weak nuclear force, electricity are all present in the accelerator as the ion passes through a charged gas and looses electrons.

In conclusion, the four fundamental forces have similar theorems, thus, scientists believe there is a single force that explains all. This is known as the Single Unified Theory, however research still continues to this day.

Lesson 9 Moving in Circles

The universe, a contantly expanding void with a plethora of spheres turning on axes. Every planet and satelite in the universe was once believed to orbit in uniform circular motion, a motion that has both constant speed and constant acceleration.

Plato, a student of Socrates, a Greek philosopher, and a mathematician stated that all motions in the heavens must be circular and uniform. This idea fell in the Greek ideal of a perfect or "ideal" universe where all bodies came to rest and had distinct and "perfect" paths of motion. Plato lived from 423 BC to 347 BC, and after this period, astronomers spent 2,000 years trying to prove Plato's findings. Early astronomers used the circular motions of the heavenly bodies to develop Astrology (reading of horoscopes), to predict harvests and growing seasons in Agriculture, and to navigate. Mesopotamians, Hindus, Native Americans, and Native Americans of Peru all tried to explain the circular motions of the planets, and some of their architectural designs followed the circular form. However, at this time, astronomers still believed the Earth was located in the center of the universe. Therefore the Aristotelian model of the universe changed to the Ptolemaic model of the universe. This change was primarily due to the discovery of the epicycle, where the planets did not orbit the earth in circles, but rather, planets orbited in circles that orbited the earth:

Aristotle's Model of the Universe:

Ptolmey's Model of the Universe:

In the field of mathematics, it was discovered that any point on a circle can be found by its Cartesian coordinates because a vector drawn to each point on a circle has the same length:

x/r = theta

y/r = theta

x = rcos(theta)

y = rsin(theta)

With the information above, each heavenly body was to be a point on a circle moving at a constant rate. The angle theta is the angle measure between the inital starting point and time (t). Therefore, [d(theta)/dt] is constant, and equivalent to omega. Thus at any time (t), theta = omega(t). After one revolution, the angle theta is 2(pi). Finally, angular speed is described by 2(pi)/T = omega. The video below displays how circular motion has two components. These components consist of circular motion and vertical motion. The vertical motion is evident when the orbiting green sphere is displayed on its side:

Circular motion is described by the equation:

r = rcos(omega (t))i^ + rsin(omega (t))j^

where both the Cartesian location [rcos(theta)i^ + rsin(theta)j^] and angular motion at any time (t) where [theta = omega(t)] are considered.

With this discovery and application, the Ptolemaic model of the universe was replaced by the Copernican system with the sun at the center of the universe:

This model explained why the planets varied in brightness and why the retrograde motion of the planets was due to the faster motion of planets with smaller orbits, but it still incorrectly portrayed the orbits of the planets in terms of uniform circular motion. Below is a video displaying the apparent retrograde motion of Mars:


Newton's Theory of the Moon proved the Copernican model and that the planets orbited the sun:

The gravity of the earth makes the moon accelerate as F = ma , the moon continues to accelerate all the time but moves at a constant speed. This contradicts the idea of the derivative where acceleration is the rate of change of velocity. Therefore, Newton realized that the velocity of the moon is actually a vector, or the distance the moon would travel if it were not under the force of the gravity of the earth. Thus, velocity is the length an object can accelerate with constant speed as long as it changes direction. This is because the derivative of a vector explains how a vector is changing either by magnitude or direction. In the case of the moon, the acceleration and velocity have constant magnitude but change direction.

The size of the vectors = distance/time
v = 2(pi)r / T

radius [ v = omega(r)]
velocity [a = omega(v)]
acceleration [a = v^2/r]

Therefore, centripital acceleration is the force required to produce constant uniform motion of a body in orbit under the force of gravity. The length of the radius vector determines gravitational force, and this force determines acceleration. This is why planets that are located farther away from the sun take a longer period of time to complete their orbits.

Circular motion requires this relationship between the magnitude of the radius, speed, and acceleration:

a = Gm(e)
          r^2

If a body starts out with just the right velocity, speed is constant and uniform circular motion results:

 a = v^2
        r
a = Gm(e)
        r^2

v^2 = Gm(e)
   r          r^2

v = sqrt(Gm(e)/r)

All in all, the planets veritably orbit the sun in elliptical paths. However, in uniform circular motion, the velocity vector is perpendicular to the radius vector  and the acceleration is inward along the radius. According to Newton's second law, any body that executes uniform circular motion  is being accelerated  by some external force. For a body in orbit, this force is gravity.

Lesson 8 The Apple and the Moon

Isaac Newton's Law of Univeral Gravitation explains why all bodies near the surface of the earth fall with constant acceleration and (in context) why an apple but not the moon falls from the sky.

Early Greek astronomers attempted to explain the motion of the bodies in space through uniform circular motion. Copernicus built upon the Greeks studies and placed the sun, rather than the earth, at the center of the "universe" or solar system. However, Copernicus still followed the idea of a uniform universe. Galileo then studied how bodies act under the influence of earth's gravity on earth. Utilizing all of these discoveries, Kepler developed three laws explaining the planetary motions "outside" of the earth in space:

First Law:

Second Law:

The closer a body is to the sun, the faster it moves and the farther a body is from the sun, the slower it moves.

Third Law:

The larger a body's orbit, the longer period of time it takes for the body to complete a revolution along its orbit.

Here is a video desbribing Kepler's three laws:


From these laws Newton deduced that the reason the planets orbit the sun at a constant rate and path, that the reason all bodies fall at the same rate, and that the reason that the moon orbits the earth were all related. Newton found that every two bits of matter in the universe attract each other and that the force of attraction is proportional to each of their masses. This force of attraction has a concentrated net force in the downward position, corresponding to the masses of the two objects. The force weakens inversely as the square of the distance between the two bodies.

F = -G m1m2 r^
               r^2
Let's consider and apple falling from a tree on earth.

F = -G m(apple)m(earth) r^      
              r^2
r = square of distance from the apple to the center of the earth.

F = m(a)a

m(a)a = -G m(apple)m(earth) r^
                               r^2
a = -Gm(E)
       R(E)^2

Gravity does not depend on the mass of the apple, thus gravity has the same effect on all bodies near earth. The constant force of gravity is evident in this video below when two objects are dropped in a vacuum:


g = -Gm(E)    which is 32ft/sec/sec
        R(E)^2

On the moon, the acceleration may be different but it still relates to the universal gravitational constant. The acceleration of a falling body on the moon is 1/6 the acceleration of a falling body on earth. However, why doesn't the moon fall from the sky like the apple?

If a projectile is thrown fast enough it will orbit the earth. Therefore, just as astronauts are held in orbit of the moon, gravity holds the moon in orbit of the earth. 0g or weightlessness is not the absence of gravity; objects are held in orbit as they "fall for eternity." The rate at which the moon falls is much smaller than the rate at which  the apple falls. In fact, the moon falls slower than the apple by a factor of 3,600. Therefore the moon falls 1/20 of an inch every second:

Moon falls towards earth

a(m) = G M(E) mass of earth
                r^2(m) square distance to earth
g = G M(E) mass of earth
          R^2(E) radius of earth
am/g = GM(E)/r^2(m)
            GM(E)/R^2(E)
am/g = R^2(E)
             r^2(m)
am/g = (R(E))2
              r(m)

Every month the moon travels 2(pi)r(m)

2(pi)r(m) = speed
1month

The Law of Inertia states that the moon wants to fly out of the orbit of the earth in a straight line, however, it is held in by the force of gravity.

The speed of the moon is 40,281 in./sec

Using the Pythagorean theorem with the orbit of the moon:

r(m)^2 + d^2 = (s(m) + r(m))^2

d^2 = [s(m)^2] + 2r(m)s(m)       [s(m)^2] is so small it is ignored.

d^2 = 2r(m)s(m)

d^2 = s(m)
2r(m)

s(m) = 1/20 in. (The rate at which the moon falls towards the earth as it "falls for eternity").



         

 

Lesson 7 Integration

In the Seventeenth Century, the idea of integration was discovered by both Leibniz and Newton. However, Newton took complete credit for this discovery and used it to find the area of curved figures. Newton took the first steps to define calculus as a mechanical calculator of both theory and practice. In ancient time periods, Greeks did not have the luxury of utilizing calculus. Instead, Greeks would perform the method of exhaustion. This method divided curved figures into a plethora of simple polygons in order to find the area of curved figures. In fact, this is how the value of pi was discovered as 3.1415... Kepler was able to discover the area and volume of ninety-two new curved shapes using this method. Below is a polygon that is divided into smaller polygons and a circle divided into smaller polygons by the method of exhaustion:

However, what if the area of the space between the curve y = x^2 for all values [0, t] and the x-axis needed to be found? The area can still be divided into rectangles and added together in order to find the area. Leibniz developed a notation for this process using the sumation symbol, therefore:

         t                                       t

summation [f(x) delta x] = integral [f(x) dx]

         a                                       a

Using integration directly the area from 0 to t can be caculated more efficiently. And once calulated, this area would be equal to the integral of y=x^2 at t subtracted by the integral of y=x^2 at 0.

 Once observed, Integration and Differentiation are inverse processes:

f(x)

            t

integral [f(x) dx]

            0

               t

d integral [f(x) dx]

dt            0

                                                                                                   t

This is the First Fundamental Theorem of Calculus:   d integral [f(x) dx]=f(t)

                                                                                      dt            0

When dealing with area (as displayed above), the area between the curve and the x-axis on the interval [t,a], is equivalent to the expression below:

                      t

A(t) = integral dA dx + A(a)

                      a dx

This is the Second Fundamental Theorem of Calculus. Here is a video describing the use of the Second Fundamental Theorem of Calculus:

Integration is not only used to find areas of curved figures but is also used in mechanics. And identical to derviatives, acceleration, velocity, and displacement are all related by their integrals.