This blog has not only required extensive amounts of time, but has opened many outlets for me to expand my knowledge in the study of physics. Both this acquired knowledge and blog will serve as a resource I can easily access during my four years at WPI (Worcester Polytechnic Institute). Thank you, Mr. Connors, for providing me with this unique opportunity. College...here I come!
Philippe Kelley - Northbridge High School class of 2012
Friday, May 18, 2012
Lesson 46 The Engine of Nature
The invention of the steam engine led to the expansion of western culture due to both transportation and economic benefits.
James Watt used a condenser to cool steam outside of the engine allowed for further developments and progress.
A cylinder with a moveable end, pushed by pressurized steam, opening a valve to the boiler emits high pressure steam that pushes the piston out doing work on the fly wheel, which pushes the cylinder back in again for the next cycle.
The water wheel led to the steam engine, where water or steam can only flow from a higher plane to a lower plane. Sadi Carnot theorized the steam engine must be comprised of mechinisms that allow for the flow of heat to progress form high coloric to low coloric.
No machine or combination of machines can ever have the effect of making more heat run up to high temperature than down to low temperature. This is known as the second law of thermodynamics.
The most effective engine is one that can transform low temperatures to high temperatures.
Isothermal - as heat is applied, gas expands and work is done. The cylinder is heated prior to the application of more heat. This will allow for a reversable action as heat may flow in both directions.
Adiabatic - a conversion that occurs without input or release of heat within a system.
In a Carnot engine, the engine follows a process that is repeated:
Isothermal - the cylinder is heated prior to the application of more heat which allows for the expansion of the gas.
Adiabatic - no heat is applied as the gas continues to expand as the heat of the cylinder continues to heat the gas.
Isothermal - the cylinder is cooled prior to the application of low temperature which allows for the gas to contract.
Adiabatic - the cylinder is not cooled as the cooled cylinder causes the gas and piston to contract nearly completely.
Gases need space to expand and contract, and require a container for heat to be released.
Carnot engine:
e (efficiency) = W/Qi
The most efficient engine that nature allows follows the principle Qo/Qi = To/Ti. These discoveries led to the future concept of entropy.
James Watt used a condenser to cool steam outside of the engine allowed for further developments and progress.
A cylinder with a moveable end, pushed by pressurized steam, opening a valve to the boiler emits high pressure steam that pushes the piston out doing work on the fly wheel, which pushes the cylinder back in again for the next cycle.
The water wheel led to the steam engine, where water or steam can only flow from a higher plane to a lower plane. Sadi Carnot theorized the steam engine must be comprised of mechinisms that allow for the flow of heat to progress form high coloric to low coloric.
No machine or combination of machines can ever have the effect of making more heat run up to high temperature than down to low temperature. This is known as the second law of thermodynamics.
The most effective engine is one that can transform low temperatures to high temperatures.
Isothermal - as heat is applied, gas expands and work is done. The cylinder is heated prior to the application of more heat. This will allow for a reversable action as heat may flow in both directions.
Adiabatic - a conversion that occurs without input or release of heat within a system.
In a Carnot engine, the engine follows a process that is repeated:
Isothermal - the cylinder is heated prior to the application of more heat which allows for the expansion of the gas.
Adiabatic - no heat is applied as the gas continues to expand as the heat of the cylinder continues to heat the gas.
Isothermal - the cylinder is cooled prior to the application of low temperature which allows for the gas to contract.
Adiabatic - the cylinder is not cooled as the cooled cylinder causes the gas and piston to contract nearly completely.
Gases need space to expand and contract, and require a container for heat to be released.
Carnot engine:
e (efficiency) = W/Qi
The most efficient engine that nature allows follows the principle Qo/Qi = To/Ti. These discoveries led to the future concept of entropy.
Lesson 45 Temperature and Gas Laws
Heat is the random motion of atoms and particles.
Temperature can only be measured in terms of its effects.
0 degrees celsius - water freezes
100 degrees celsius - water boils
32 degrees fahrenheit - water freezes
212 degrees fahrenheit - water boils
Temperature also affects pressure. Pressure is the force per unit area exerted.
Pressure = force/area
A molecule changes momentum when it exerts force on the wall of a balloon. The opposite forces results in the expansions of a balloon.
Heating a gas causes an increase in kinetic energy of the molecules, resulting in more pressure. The pressure of a gas is proportional to the number of molecules, inversely proportional to the volume, and proportional to the average kinetic energy of the molecule. The constant of proportionality is 2/3.
p = 2/3 (N/V) k
pv = constant if temperature is constant
p1v1 = p2v2 Boyle's Law
_
constant = 2/3 Nk
pv is proportional to the kinetic energy of all molecules of the gas. This kinetic energy is a form of heat. Heating a gas causes pressure to rise and volume to rise.
All gasses expand by the same amount with a given rise in temperature. At a given pressure the volume of gas changes by the same fraction for each degrees in temperature.
v1/t1 = v2/t2 Charles's Law
There exists a temperature so low that a gas would fill no volume. This temperature is known as absolute zero, where a gas has no heat.
Absolute zero is -273 degrees celsius, -459 degrees fahrenheit, and 0 degrees kelvin.
pv = knt determines the size of one kelvin where k = 1.38 x 10^-23 J/K
Kinetic Theory of Gasses
The kinetic energy of a gas, or the collective effects of molecular collisions, is what gives a gas pressure and volume.
Temperature and heat connection:
_
knt = (2/3) Nk
pv = knt
_
pv = (2/3) nk
_
kt = (2/3) k - absolute temperature
Heat and temperature are related by kinetic energy of the molecules of gas.
Heat in a gas is the average kinetic energy of molecules.
Absolute temperature is (2/3) the mean average kinetic energy of one molecule of gas
pv = Nkt - all temperature and pressure are related by the ideal gas law, which leads to the kinetic theory.
Temperature can only be measured in terms of its effects.
0 degrees celsius - water freezes
100 degrees celsius - water boils
32 degrees fahrenheit - water freezes
212 degrees fahrenheit - water boils
Temperature also affects pressure. Pressure is the force per unit area exerted.
Pressure = force/area
A molecule changes momentum when it exerts force on the wall of a balloon. The opposite forces results in the expansions of a balloon.
Heating a gas causes an increase in kinetic energy of the molecules, resulting in more pressure. The pressure of a gas is proportional to the number of molecules, inversely proportional to the volume, and proportional to the average kinetic energy of the molecule. The constant of proportionality is 2/3.
p = 2/3 (N/V) k
pv = constant if temperature is constant
p1v1 = p2v2 Boyle's Law
_
constant = 2/3 Nk
pv is proportional to the kinetic energy of all molecules of the gas. This kinetic energy is a form of heat. Heating a gas causes pressure to rise and volume to rise.
All gasses expand by the same amount with a given rise in temperature. At a given pressure the volume of gas changes by the same fraction for each degrees in temperature.
v1/t1 = v2/t2 Charles's Law
There exists a temperature so low that a gas would fill no volume. This temperature is known as absolute zero, where a gas has no heat.
Absolute zero is -273 degrees celsius, -459 degrees fahrenheit, and 0 degrees kelvin.
pv = knt determines the size of one kelvin where k = 1.38 x 10^-23 J/K
Kinetic Theory of Gasses
The kinetic energy of a gas, or the collective effects of molecular collisions, is what gives a gas pressure and volume.
Temperature and heat connection:
_
knt = (2/3) Nk
pv = knt
_
pv = (2/3) nk
_
kt = (2/3) k - absolute temperature
Heat and temperature are related by kinetic energy of the molecules of gas.
Heat in a gas is the average kinetic energy of molecules.
Absolute temperature is (2/3) the mean average kinetic energy of one molecule of gas
pv = Nkt - all temperature and pressure are related by the ideal gas law, which leads to the kinetic theory.
Lesson 44 Energy, Momentum, and Mass
Einstein developed the Theory of Relativity, to which momentum also applies.
A ball will not come to rest until something stops it.
Po = P1 + P2
Po^2 = P1^2 + P2^2
If one ball is at relative rest:
Po = P2
Po^2 = P2^2
When objects collide the change in force is equal and opposite, so is momentum.
p = mv
F = d/dt(p)
F1 = -F2
dp1/dt = -dp2/dt
d/dt (p1 + p2) = 0 the total momentum of both balls remains constant.
Momentum remains contant in Einstein's Theory of Relativity.
In outer space, the momentum of two colliding objects released while in motion is equal and opposite as on earth.
M delta(U) = delta(P)
However, according to the theory of relativity, a collision of two balls occurs and results in equal momentum where less velocity is accounted for with greater mass. This increase in mass is influenced by speed.
F = ma
Mass is an object's resistance to change in velocity.
According to the Theory of Relativity, the mass of a moving object depends on how fast it is moving. The mass m is equal to mo (rest mass) x gamma.
Mass, like time and distance, depends on one's point of view. An object at rest resists less than object in motion. Mass depends on speed, but no mass can reach the speed of light. Inside any accelerator electromagnetic fields exert force on a tiny ion. This force increases the ion's momentum mo(v), so the speed increases and so does the mass. As objects become more massive, it becomes more difficult to make them accelerate. The ion's momentum and energy continue to increase but the speed never reaches the speed of light.
Increasing the kinetic energy of a bod increases its mass.
x1
W = integral (F) dx
xo
Kinetic energy or force changes momentum.
K = integral dx/dt dp p = mv
= integral v dp dp = mo (1/(1-v^2/c^2)) dv
K = integral v mo (1/(1-v^2/c^2))^(3/2) dv
K = mo (c^2/(1-v^2/c^2)^(1/2)) - moc^2
K = (m - mo) c^2
Eo = moc^2 - rest mass energy
E = mc^2 - total energy of the body
A loss in energy accounts for a loss in mass.
A ball will not come to rest until something stops it.
Po = P1 + P2
Po^2 = P1^2 + P2^2
If one ball is at relative rest:
Po = P2
Po^2 = P2^2
When objects collide the change in force is equal and opposite, so is momentum.
p = mv
F = d/dt(p)
F1 = -F2
dp1/dt = -dp2/dt
d/dt (p1 + p2) = 0 the total momentum of both balls remains constant.
Momentum remains contant in Einstein's Theory of Relativity.
In outer space, the momentum of two colliding objects released while in motion is equal and opposite as on earth.
M delta(U) = delta(P)
However, according to the theory of relativity, a collision of two balls occurs and results in equal momentum where less velocity is accounted for with greater mass. This increase in mass is influenced by speed.
F = ma
Mass is an object's resistance to change in velocity.
According to the Theory of Relativity, the mass of a moving object depends on how fast it is moving. The mass m is equal to mo (rest mass) x gamma.
Mass, like time and distance, depends on one's point of view. An object at rest resists less than object in motion. Mass depends on speed, but no mass can reach the speed of light. Inside any accelerator electromagnetic fields exert force on a tiny ion. This force increases the ion's momentum mo(v), so the speed increases and so does the mass. As objects become more massive, it becomes more difficult to make them accelerate. The ion's momentum and energy continue to increase but the speed never reaches the speed of light.
Increasing the kinetic energy of a bod increases its mass.
x1
W = integral (F) dx
xo
Kinetic energy or force changes momentum.
K = integral dx/dt dp p = mv
= integral v dp dp = mo (1/(1-v^2/c^2)) dv
K = integral v mo (1/(1-v^2/c^2))^(3/2) dv
K = mo (c^2/(1-v^2/c^2)^(1/2)) - moc^2
K = (m - mo) c^2
Eo = moc^2 - rest mass energy
E = mc^2 - total energy of the body
A loss in energy accounts for a loss in mass.
Lesson 43 Velocity and Time
If one uses a loop of wire and a galvanometer a light will move when an electric current passes through the wire. This electric current can be created by moving and removing a bar magnet from the loop of wire. The force that sets electric charges in a wire in to motion are described by the equation below:
F = qE + qv x B
Einstein desired to further explain why this equation remains true and developed the Theory of Relativity and found that time is relative:
delta(t) = gamma delta (t)o
gamma = 1/sqrt(t - v^2/c^2)
According to the Theory of Relativity, timed events occur more slowly by a factor of gamma, depending on the position of the observer. The Theory of Relativity is a theory of motion based upon different observers where there is no absolute motion and no absolute rest.
Nothing travels faster than the speed of light. If an observer is detecting the speed of a baseball thrown from a moving platform the speed or velocity cn be determined by the slope of the observer and baseball in the space-time continuum where both space and time are independent axes.
v' = delta(x)'/delta(t)'
Speeds computed in the spacetime frame are always less than one, or the speed of light.
Ux = (U'x + v)/(1+ VU'x/c^2)
Even along the direction of relative motion, components of velocity never add to a speed greater than the speed of light.
Uy = U'y/(gamma(1 + VU'x/c^2))
Uz = U'z/(gamma(1 + VU'x/c^2))
The veocity transformation does affect components perpendicular to the motion because although distances remain the same, time changes from one frame to the next.
There is a test of relativity in the phenomenon decay of cosmic ray muons.
Because time is relative, if one travels close to the speed of light to an area of space ten lightyears away, due to Lorentz contractions, the individual travelling only ages several months. An individual on earth ages 20 years.
In a completely empty universe time and space have no relevance. In a universe with many bodies, no object is at complete rest, but remains in motion until an object acts upon it, following the law of inertia. Time and space are relevant in this universe. Rest can only be observed in frames of time and space.
F = qE + qv x B
Einstein desired to further explain why this equation remains true and developed the Theory of Relativity and found that time is relative:
delta(t) = gamma delta (t)o
gamma = 1/sqrt(t - v^2/c^2)
According to the Theory of Relativity, timed events occur more slowly by a factor of gamma, depending on the position of the observer. The Theory of Relativity is a theory of motion based upon different observers where there is no absolute motion and no absolute rest.
Nothing travels faster than the speed of light. If an observer is detecting the speed of a baseball thrown from a moving platform the speed or velocity cn be determined by the slope of the observer and baseball in the space-time continuum where both space and time are independent axes.
v' = delta(x)'/delta(t)'
Speeds computed in the spacetime frame are always less than one, or the speed of light.
Ux = (U'x + v)/(1+ VU'x/c^2)
Even along the direction of relative motion, components of velocity never add to a speed greater than the speed of light.
Uy = U'y/(gamma(1 + VU'x/c^2))
Uz = U'z/(gamma(1 + VU'x/c^2))
The veocity transformation does affect components perpendicular to the motion because although distances remain the same, time changes from one frame to the next.
There is a test of relativity in the phenomenon decay of cosmic ray muons.
Because time is relative, if one travels close to the speed of light to an area of space ten lightyears away, due to Lorentz contractions, the individual travelling only ages several months. An individual on earth ages 20 years.
In a completely empty universe time and space have no relevance. In a universe with many bodies, no object is at complete rest, but remains in motion until an object acts upon it, following the law of inertia. Time and space are relevant in this universe. Rest can only be observed in frames of time and space.
Lesson 42 The Lorentz Transformation
The Michelson-Morley experiment set out to determine the existence of the luminiferous ether, through which the speed of the earth was belived to be determined. Michelson and Morley attempted to determine the existence of the ether through its effects on the speed of light.
The reason both light beams reach the interferometer at the same time is due to the contraction of one of the device's arms. Lorentz supported this idea with the theory of the properties of electrons where electrons contract in the direction of motion. This led to the idea of relativity where the speed of an object is determined through the speed of the observer.
Time and distance are affected by motion.
The speed of light is the same for all observers. However, if light is in motion between two mirrors, the observer will see the light taking less time to complete one cycle. This is due to the increased diagonal distance.
The relativity of time is derived from the right triangle formed by the distances traveled.
The Pythagorean theorem shows that the path of the moving light is longer than the distance between the two mirrors.
delta(t) = 1/sqrt(1-(v^2/c^2))delta(t)'
delta(t) = gamma delta(t)'
Two observers can agree on the occurence of an event with respect to time using the Lorentz transformations
Galileo - x' = x - vt
Lorentz - x' = gamma( x - vt) the equation along the direction of motion.
y' = y the equation describing the perpendicular motion that is the same in both frames of observation.
z' = z
t' = gamma (t - vx/c^2)
The Lorentz transformations link time and space together.
Einstein developed two postulates:
1st - the laws of physics are the same for all inertial frames
2nd - the speed of light is the same for all observers.
The reason both light beams reach the interferometer at the same time is due to the contraction of one of the device's arms. Lorentz supported this idea with the theory of the properties of electrons where electrons contract in the direction of motion. This led to the idea of relativity where the speed of an object is determined through the speed of the observer.
Time and distance are affected by motion.
The speed of light is the same for all observers. However, if light is in motion between two mirrors, the observer will see the light taking less time to complete one cycle. This is due to the increased diagonal distance.
The relativity of time is derived from the right triangle formed by the distances traveled.
The Pythagorean theorem shows that the path of the moving light is longer than the distance between the two mirrors.
delta(t) = 1/sqrt(1-(v^2/c^2))delta(t)'
delta(t) = gamma delta(t)'
Two observers can agree on the occurence of an event with respect to time using the Lorentz transformations
Galileo - x' = x - vt
Lorentz - x' = gamma( x - vt) the equation along the direction of motion.
y' = y the equation describing the perpendicular motion that is the same in both frames of observation.
z' = z
t' = gamma (t - vx/c^2)
The Lorentz transformations link time and space together.
Einstein developed two postulates:
1st - the laws of physics are the same for all inertial frames
2nd - the speed of light is the same for all observers.
Lesson 41 The Michelson-Morley Experiment
The Michelson-Morley Experiment was designed to detect the motion of the earth through the luminiferous ether.
The ether was created to explain how light travels from the sun to the earth, as light waves propogate from the sun through this medium.
The experiment proved there was no ether, however, Einstein developed the Theory of Relativity from its failure, which explained what the experiment attempted to do.
Due to the Theory of Relativity and the failure of the Michelson-Morley experiment, space was now recognized as a vacuum.
If the ether was viscous, planets would lose energy and spiral inward towards the sun. Therefore, physicists believed the ether was a perfectly mobile fluid with no viscosity. It is virtually incompressible, transparent, and fills all of space allowing planets to obey Newton's laws with ease.
Michelson developed the interferometer that splits a light in two beams. As light travels through a partially transparent and opaque mirror, light is sent in two beams in perpendicular directions. The light is then sent back by two opaque mirrors and the light combines into one beam. If the beams of light ravel at the same speed, the interferometer produces a read-out of a spiral with a single dot in the center.
The speed of light c = 3 x 10^8 m/s
The speed of the earth through space v = 3 x 10^4 m/s
Difference in completion time (v/c)^2 = 10^-8
Lorentz developed the Lorentz transformations (mathematical equations) to discuss the phenomenon of the retraction of a mirror in the interferometer.
Henri Poincare developed the priciple of relativity, that absolute motion will never be observed in a laboratory.
The ether was created to explain how light travels from the sun to the earth, as light waves propogate from the sun through this medium.
The experiment proved there was no ether, however, Einstein developed the Theory of Relativity from its failure, which explained what the experiment attempted to do.
Due to the Theory of Relativity and the failure of the Michelson-Morley experiment, space was now recognized as a vacuum.
If the ether was viscous, planets would lose energy and spiral inward towards the sun. Therefore, physicists believed the ether was a perfectly mobile fluid with no viscosity. It is virtually incompressible, transparent, and fills all of space allowing planets to obey Newton's laws with ease.
Michelson developed the interferometer that splits a light in two beams. As light travels through a partially transparent and opaque mirror, light is sent in two beams in perpendicular directions. The light is then sent back by two opaque mirrors and the light combines into one beam. If the beams of light ravel at the same speed, the interferometer produces a read-out of a spiral with a single dot in the center.
The speed of the earth through space v = 3 x 10^4 m/s
Difference in completion time (v/c)^2 = 10^-8
Lorentz developed the Lorentz transformations (mathematical equations) to discuss the phenomenon of the retraction of a mirror in the interferometer.
Henri Poincare developed the priciple of relativity, that absolute motion will never be observed in a laboratory.
Subscribe to:
Posts (Atom)