Anthology of Ideas » Physics http://anthologyoi.com Anthology of Ideas is an archive of thoughts and form. Sat, 03 Mar 2012 11:16:55 +0000 en hourly 1 http://wordpress.org/?v=3.3.1 Water balloon not exploding in high-speed http://anthologyoi.com/physics/water-balloon-not-exploding-in-high-speed.html http://anthologyoi.com/physics/water-balloon-not-exploding-in-high-speed.html#comments Tue, 01 Apr 2008 02:22:11 +0000 aaron http://anthologyoi.com/?p=551 I am so totally stealing this video from Talk Like a Physicist, but it is worth it and I promise I won’t do it again. I really don’t have anything intelligent to add except: COOL!

]]> http://anthologyoi.com/physics/water-balloon-not-exploding-in-high-speed.html/feed 1 Calculus Based Physics Formulas: Mechanics http://anthologyoi.com/physics/calculus-based-physics-formulas-mechanics.html http://anthologyoi.com/physics/calculus-based-physics-formulas-mechanics.html#comments Fri, 23 Mar 2007 13:52:05 +0000 aaron http://anthologyoi.com/physics/calculus-based-physics-formulas-mechanics.html Continue reading →]]> This is just a basic equation list, explanations can be found elsewhere. For the most part derivations are done for you, but it is beneficial to understand how an equation goes from one form to another.

One dimensional Equations of motion (along a single vector direction)
Velocity as a function of time : v_{xf} = v_{xi} + a_x t
Position as a function of time: x_f = x_i + v_{xi}t + \frac{1}{2} a_x t^2
Velocity as a function of position: v^2_{xf} = v^2_{xi} + 2a_x ( x_f &#8211; x_i)<br />

Projectile Motion
Horizontal motion
Velocity along x: v_{xi} = v_i cos(\theta)
Position from position as a function of time: x_f = v_i cos(\theta)t
Max Horizontal dist: R = v^2_i \frac{sin( 2 \theta_i)}{g}

Vertical Motion
Velocity along y:v_{yi} = v_i sin(\theta)
Position: from position as a function of timey_f = y_i + v_{yi}t &#8211; \frac{1}{2} g*t^2
Maximum Height: h_{max} = v^2_{i} \frac{sin(\theta_i)}{2g}

Circular Motion
Radial Acc: a_r = v^2_r = a cos( \theta)
Tan. Acc:a_t = \frac{d \mid \vec {v}\mid}{dt} = a sin(\theta)= r \alpha
Total Acc (magnitude) from Pythagoras: a = \sqrt{a_r^2 + a_t^2}

The Laws of Motion
Newtons Second Law: \sum{F_{x,y, or z}} = ma_{x,y, or z}
Equilibrium Conditions: \sum {F_{x,y, or z}} = 0
Force of Static Friction F_{s max} = \mu_s*n
Force from Kinetic Friction F_{k max} = \mu_k*n

Force/Work
Constant Force: w_{net} = \vec{f_{net}}*\delta r = F * r cos(\theta) = \delta K
Variable Force: w_{net} = \int f_{net} d \vec{r}
Hooke’s Law: f_s = -k x
Spring Work: w = \frac{1}{2} k x_i^2 &#8211; \frac{1}{2} k x^2_f
Kinetic energy: k = \frac{1}{2} m v^2
Work – kinetic energy theorem: w_{net} = \delta k = k_f &#8211; k_i, k_f = k_i
Power: P = \frac {\Delta w}{\Delta t} ,p = \frac {de}{dt} , P = \vec{f} \vec{v}
gravitational potential energy: U = mgh
conservation of mechanical energy: E = K_f + U_f = K_i + U_i = const. + \mid f_k \delta x \mid
elastic collision conserved moment and KE: v_{1f} = (\frac{m_1 &#8211; m_2}{m_1 + m_2}) v_{1i} + (\frac {2 m_2}{m_1 + m_2}) v_{2i}
2d elastic (comp): m_1 v_{1ix} + m_2 v_{2ix} = m_1 v_{1fx} + m_2 v_{2fx},m_1 v_{1ix} = m_1 v_{1f} cos(\theta) + m_2 v_{2fx}cos(\phi)
KE conservation for elastic: \frac{1}{2} m_i v_i^2 +\frac{1}{2} m_{2i} v_{2i}^2 = \frac{1}{2} m_1 i v_{1f} i^2 + \frac{1}{2} m_2 i v_{2f} i^2
Momentum: \vec{P} = m \vec{{v}

Mass
Center of mass (comp): x_{cm} = \frac{\sum_{i=1}^{n} m_i x_i }{m}
Position vector for CM: \vec{r_{cm}} = x_{cm} \vec {i} + y_{cm} \vec {j} + z_{cm} \vec {k}
Continuous mas dist: x_{cm} =\frac {1}{m} \int{\lambda dx}
Mass of Uniform: m= \int{\lambda dx}
Linear Mass Dist: \lambda = \frac{m}{l} = \frac{dm}{dl}
Area Mass Dist: \omega = \frac{m}{a} = \frac{dm}{da}

Rotational Motion
angular speed: \omega = \frac{d\theta}{dt}
angular acceleration: \alpha = \frac{d\omega}{dt} ,\frac{a_t}{r},\frac{\tau}{I}
Moment of Inertia: I = m_i r_i^2,I = \int r^2 dm,I = \int (density)r^2 dv,I = I_cm + mD^2
Rotational KE: K_R = \frac{1}{2} I \omega^2
Work: \frac{1}{2} I \omega_f^2 &#8211; \frac{1}{2} I \omega_i^2
Net torque : \sum\tau = I \alpha, \sum \tau =\frac {dL}{dt}
Work : W = \int_{\theta_f}^{\theta_i} t d\theta
Power : P = \tau \omega
Angular Momentum : L = I \omega
Torque: : \tau = rF sin(\theta)

Moments of Inertia
Hoop : I_{cm} =mr^2
Cylinder (hollow) : I_{cm} = \frac{1} {2} m(r_1^2 + r_2^2)
Cylinder : I_{cm} = \frac{1} {2} mr^2
Rectangular Plate : I_{cm} = \frac{1} {12} m(a^2 + b^2)
Rod (center rotate): I_{cm} =\frac{1} {12} mL^2
Rod (end Rotate): I_{cm} =\frac{1} {3} mL^2
Solid Sphere : I_{cm} = \frac{2} {5} mr^2
Spherical Shell : I_{cm} = \frac{2} {3} mr^2

]]> http://anthologyoi.com/physics/calculus-based-physics-formulas-mechanics.html/feed 7 Units in Physics (mechanical, electricity, magnetism, light and optics) including Si units. http://anthologyoi.com/physics/units-in-physics-mechanical-eletricity-magnetism-light-and-optics-including-si-units.html http://anthologyoi.com/physics/units-in-physics-mechanical-eletricity-magnetism-light-and-optics-including-si-units.html#comments Wed, 28 Feb 2007 00:11:18 +0000 aaron http://anthologyoi.com/physics/units-in-physics-mechanical-eletricity-magnetism-light-and-optics-including-si-units.html Continue reading →]]> This is a reference list with notes of all SI and derived units in physics. The notes provide a brief explanation of some of the more confusing elements, but be warned that the full explanation could take many pages, and may be explained elsewhere on this website.

Physics has only 5 base units. (Plus the SI units Mole and Candela, but these are rarely used in Physics.)

Name Abbreviation (Symbol) Standard Unit Notes
Name Abbreviation (Symbol) Standard Unit Notes
Length l, x (for distances) Meter (m) A meter is defined as the distance light travels in a vacumm in \frac{1}{299 792 458} of a second (in physics it is customary to use metric measurements although the basic principles apply if you to use feet instead of meters)
Mass m, M (when used with measurements in meters) Kilogram (kg) A kilogram is defined as the weight of a specific platinum-iridium cylinder
Time t Second (s) Seconds are defined as 9,192,631,770 vibrations of radiation from a cesium atom
Temparature T Kelvin (K) A degree kelvin is defined as \frac{1}{273.16} of the distance between absolute 0 and the triple point of water
Electric Current I Ampere (A) An ampere is the amount of charge (C) passing through a surface per second, and is defined as the current which produces a force of 2*10^{-7} newtons per meter of length between two infinitely long, perfectly straight and parallel conductors with an infinitely small cross section separated by one meter in a vacuum..

Each of these base units is defined on fundamental constants, and all other units are based on these five units. At times it useful to break longer equations down to their most basic units to determine if the equation makes sense. The most common combinations of these basic units are given their own symbols and names. These common units are as follows.

Name (alphabetically) Abbreviation (Symbol) Unit – Derivation Notes
Name (alphabetically) Abbreviation (Symbol) Unit – Derivation Notes
Acceleration a \frac{\text{m}}{\text{s}^2} (meters per second squared) Acceleration is literally the rate of change of the rate of change of an object’s position.
Angle \theta,\varphi radian A radian is defined as the angle an arc length, equal to the circle’s radius, makes with the center of the circle.
Capacitance C Farad (F) \frac{\text{C}}{\text{V}} (Charge over the Potential)
Charge Q ), q ,e (of elementary particles) Coulomb (C) \text{A*s} (Amperes times seconds) Literally the charge is the amount of current that flows over the entire time period.
Density \rho \frac{\text{kg}}{\text{m}^3} Density is the amount of mass in every cubed unit length
Displacement s, d (distance), h (height) meters – m Displacement is the total change in length in any single direction. Sometimes it is used as the absolute change in distance –if you were walk all the way around the earth less one meter, your displacement would be one meter
Electric Field E \frac{\text{V}}{\text{m}} (Electric potential per meter) An electric field is the amount of electric potential over any given area.
Electric Flux \Phi_e \text{V*M}(Electric Potential time meters) Electric flux is the electric field through some area.
Electromotive Force (emf) \scr{E} or \epsilon Volt (V) Electromotive force is a potential difference in volts.
Electron Volt eV \text{e*J} The Electron Volt is the amount of energy change of a charge-field system when a charge of magnitude e is moved through a potential difference of 1V. It is used in place of the Joule (energy).
Energy E (total),U (potential),K (kinetic) Joule (J) kg * \frac{\text{m}^2}{\text{s}^2}
Entropy S \frac{\text{J}}{\text{K}}
Force F Newton (N) kg \frac{\text{m}}{\text{s}^2}, \frac{\text{J}}{\text{m}}
Frequency f,v Hertz (Hz) \frac{\text{cycles}}{\text{s}}
Heat Q Joule (J) Joule is used for work, heat and energy, but remember that Joule is a unit of energy not of energy transfer. This means that heat can have a Joule of energy, but can’t be measured in Newtons.
Magnetic Field B Telsa (T) \frac{\text{Wb}}{\text{m}^2}
Magnetic Flux \Phi_m Weber (Wb) \text{kg} \frac{\text{m}^2}{\text{A}} \text{s}^2
Momentum p \text{kg} \frac{\text{m}}{\text{s}}
Potential (Electric) V or \delta V Voltage (V) \frac{\text{J}}{\text{C}} Potential is the amount of excess charge. It can be compared with potential energy.
Power P, \scrP \frac{\text{j}}{\text{s}} Power is the amount of work done in any given time.
Pressure P Pascal (Pa) \frac{\text{N}}{\text{m}^2}
Resistance R Ohm \frac{\text{V}}{\text{A}} Resistance is the amount of energy that is lost in the transfer of energy through an object.
Torque \tau \text{N*m} Torque is usually shown with units N*m even though it is technically joules.
Velocity v \frac{\text{m}}{\text{s}} Velocity is the speed and direction of an object.
Wavelength \lambda Meter (m)
Work W Joule (J) \text{N*m} Work is the amount of force outputted over some distance.

For of the derived units such as electric flux there are multiple possible unit definitions. The two standard definitions for electric flux are Vm and \frac{Nm^2}{C} as the following derivation from the former to the latter shows they are both correct.

V*m = \frac{J}{C}m = \frac{\frac{kg*m^2}{s^2}}{As}*m = \frac{kg*m^3}{As^3} = \frac{Nm^2}{As} = \frac{Nm^2}{C}

This derivation only used units given in the above table, but there are other ways.

]]> http://anthologyoi.com/physics/units-in-physics-mechanical-eletricity-magnetism-light-and-optics-including-si-units.html/feed 23 Aesopian Fable for Modern Times http://anthologyoi.com/writings/fiction-aesopian-fable-for-modern-times.html http://anthologyoi.com/writings/fiction-aesopian-fable-for-modern-times.html#comments Fri, 28 Jul 2006 22:21:54 +0000 aaron http://localhost/anthologyoi/papers/fiction-aesopian-fable-for-modern-times.html Continue reading →]]> The First Law of Thermodynamics is: “Energy cannot be created or destroyed; it may be transformed form one form into another, but the total amount of energy never changes.” This law can be translated to say that because energy is finite then matter also is finite.
The Second Law of Thermodynamics is: “in all energy exchanges some energy must be used to transfer the energy from an object with less energy to one with more energy.” This law can also be taken to mean that energy is used as energy is transferred.
Without most people even realizing it these two laws rule the world around us. The strive for energy and heat has caused wars between peoples of all technology levels. For without it few can survive. In our modern world oil is the energy producer of choice but oil will run out, energy trapped in its particles will be released back into the universe to serve as a different type of energy. For that is the fate of all things. Nothing ends and nothing begins; forms may change but everything is made out of pieces of the things that came before. Energy does not replenish itself, like water it can be used too quickly or too wastefully. What would happen if humankind achieved the ultimate goal, the harnessing of all energy, to have it at their command? Would it bring happiness or sorrow? Would it bring pleasure or pain? Could humans even be trusted with the possibility? There are no definitive answers, just dreams and fantasies.
Think of the world around you, cars number in the millions each consuming far more fuel than they need, houses are built larger and larger, wars are fought over oil and forests strips for firewood. Many of the conflicts of today are based on the scarcity of resources: oil, food and land. The energy that is produced from the burning of coal, firewood and oil is not free, nature spent many years transferring solar energy into sugars to grow the trees, animals spent energy eating the trees to eventually die and become oil, and the earth itself used energy to heat and cool magma in such a way as to create coal. Unfortunately this process thanks to the second law is always run at a deficit and humankind continues to exploit these resources without concern over the future well-being of the planet or humankind.
Now imagine a world, much like this one, where uranium, oil, coal and wood are no longer used for energy. The fight over combustion has ended, the ultimate form of energy harnessing has been perfected, there are no longer conflicts over resources, every man, woman and child can acquire exactly what they need with little in the way of work. This new energy creation is not the forms of the past, no windmills or solar plants are used. Instead the very energy of the universe is harnessed to manipulate both matter and energy to change it into the petty desires of the common man. Soldiers no longer fight wars, farmers no longer plant crops, business men no longer do business, in fact no one does anymore than is required. The sick can heal themselves, the hungry can feed themselves, there is no need for any thought outside temporary pleasure.
This idea may seem pleasurable to the workers, the sick or the hungry, but the surface image of an idyllic yet under-preforming world hides a much bleaker image. As the machines that harnessed the previously non-exploitable energy of the universe, global warming stopped. Those who argued that combustion was the cause of the global warming declared victory, but the average man did not notice but instead just turned the heat up. As the years went by the days got shorter, but the average man just turned the lights on longer. Centuries pass, it now seems to be an endless winter night, the days are long and cold, the earth trembles at its core no longer able to rejuvenate itself.
Even the seemingly unlimited energy of the universe has a limit, energy can not be destroyed or created, but can be wasted. For the people of the above world they thought they found an unlimited supply of energy, but everything has a limit. As they stuffed themselves with food, lengthened their lives and built houses they manipulated the energy of everything around them. Energy flows from one point to another and every link in the chain is effected when one part changes its circumstances. The energy the people used was not from some mystic energy filled place in the universe that would never run dry but instead it can from themselves, their planet and their sun. The free use of resources met with disaster because they believed themselves above the mortal chains of the universe, because they forgot the most basic laws of the universe.
One may read this little fantasy and disregard it as an exaggeration of reality, but this is unwise. Humans in the current time show a great disregard for the resources around them, energy is wasted day and night by people from many different societies. The two laws show us that no man is an island, what one person does effects all other people and only through careful stewardship can the energy around us be harnessed in a manner that is not wasteful. Otherwise in the future man may become like the above world, lazy and killing themselves with their own pitiful desires.

]]> http://anthologyoi.com/writings/fiction-aesopian-fable-for-modern-times.html/feed 0 Using the Rules of Physics to Find the Mass of the Planet Crouching Tiger, Hidden Dragon Takes Place on. http://anthologyoi.com/physics/physics-using-the-rules-of-physics-to-find-the-mass-of-the-planet-crouching-tiger-hidden-dragon-takes-place-on.html http://anthologyoi.com/physics/physics-using-the-rules-of-physics-to-find-the-mass-of-the-planet-crouching-tiger-hidden-dragon-takes-place-on.html#comments Sat, 24 Jun 2006 22:13:59 +0000 aaron http://localhost/anthologyoi/papers/physics-using-the-rules-of-physics-to-find-the-mass-of-the-planet-crouching-tiger-hidden-dragon-takes-place-on.html Continue reading →]]> The clip that is referenced is near the end of the movie after the two female characters fight at the school, and runs for about 2 -3 minutes.

It doesn’t take any physics experience to realize that “Crouching Tiger Hidden Dragon” is a bad physics movie, however with physics we can show just how bad it really is. The clip I selected runs only a few seconds long but portrays many of the formula’s that one would use in a simple mechanics physics course. Through careful calculations and a little estimation (all double checked) I was able to determine a range of details including the acceleration of gravity on the “planet” this movie takes place in, the diameter of the planet, and the density of the planet. I was also able to determine a few specific examples aside from the acceleration of gravity that show that this movie is a BPM.

To calculate the acceleration of gravity I used two different scenes, the first scene used was from the beginning of the clip where the actress jumped a single full jump over part of a body of water. By finding the actresses real height I estimated the following ratio of 1.22 meters = 1cm (second trial 1cm = 1.5m). I then calculated the ground distance of her jump to be about 6 m (second trial 9m) long and her angle of jump to be 50 degrees, using these two pieces of information i found the actual distance she covered in her arc to be 9.48 meters (second trial 14m). The time of this jump is roughly 3 seconds long. Using a grapher on my calculator I graphed a curve of her jump (trial one) that started at (0,0) reached a maximum of (1.5, 1.4) and (3,0) the equation of this curve (basic height function) is: F(x) = .8124x^2 +2.437x , where x is the time and F(x) is the distance above the ground. I then found the second derivative of the function for an instantaneous acceleration of 1.6248 m/s^2, however actual distances predict she should have been traveling at 2.5 m/s^2. Because all the numbers were estimated it is easy to see how a little bit can make a big difference, in this case by increasing trial ones height to 2m and decreasing the time the jump takes to 2.6 seconds (both of which are easily in the margin of error) we get an acceleration of 2.45 m/s^2. This whole process was then repeated with a larger scale and ended with an instantaneous acceleration of 2.5m/s^2 confirming the earlier number because the distances are roughly double.

Obviously this is a BPM because acceleration of a falling object is 9.8 m/s^2, had she been falling at this speed she would have traversed 11m instead of just under 9.48m.

This same result is seen in the second example scene where she “falls”� onto the pond. Again using her height as the base we find she fell about 12m down in three seconds, which when plugged into the distance formula gives us the acceleration we were expecting from the above (2.5 m/s^2).
The third example is using the gravity constant and the above acceleration to find the mass of the planet and the distance from the center. Starting with the formula: 2.5 = 6.67×10^-11 (M/d^2) where 2.5 is the acceleration and M is the planets weight in kilograms. Solving the equation for “d” we find that d = .000005*m1/2 (Solving for M gives us 3.7 *1010*d^2) unfortunately we only have a single equation so it is impossible to solve for a single number M in the prior equation, however we can use another M as a base and then solve.

The “D” we have is an exponential function that gives the Distance from the center of a planet with mass of “M”. Therefore we now have a formula that can calculate the information of any planet with mass M (or distance from the same) that has an acceleration from gravity of 2.5 m/s^2.
Volume = .75*pi*(.000005*m^½)^3
Density = M/ .75*pi*(.000005*m^½ )^3
Circumference = 2*pi* (.000005*m^ ½ )
For example:
To if we use earth as the mass we find that we are 1.2*107m away from earth ( this number puts us outside the moon’s orbit), that the volume of the space closer to earth is 4.3*10^21 m^3, the density is 1386.1 kg /m and the circumference of the space is 7.6*107m.
Comparing those numbers with the real ones:
Distance from the center on the surface of the earth: 6378.1 km
Earths volume: 1.0832�?1012 km^3
Density: 5,515 kg/m³
Circumference: 40,075. km
We find that to have the acceleration of earths gravity at 2.5 m/s^2 the actors would have to have been 1881.44 times as far away from the center of earth than if they had had 9.8 m/s^2 as their acceleration. Using the rule of inverse squares the actress with mass 42kg has a weight of 411.6n on earth but would only weigh 1/3545689th (.000116n) as much as if she was on earth.
Of course there are many many other examples in the same scene that can be used, but they all focus on the same things.

]]> http://anthologyoi.com/physics/physics-using-the-rules-of-physics-to-find-the-mass-of-the-planet-crouching-tiger-hidden-dragon-takes-place-on.html/feed 3 Einstein’s Dreams by Alan Lightman http://anthologyoi.com/writings/books/literature/einsteins-dreams-by-alan-lightman.html http://anthologyoi.com/writings/books/literature/einsteins-dreams-by-alan-lightman.html#comments Sat, 24 Jun 2006 06:20:41 +0000 aaron http://localhost/wordpress/?p=8 Continue reading →]]> This book asks the questions what would Einstein’s dreams be like leading up to the publishing of the Theory of Relativity. Lightman is kind enough to give us quite a few short stories that are written in a powerful and insightful way.
The stories includes scenarios such as what would happen if time ran at different speeds in different towns. Although the stories on the surface sound fanciful and utter rubbish they dig deep into modern society and in a very subtle way critique they way people live their lives. Lightman also brings up many of the doubts people have in their lives and portray both the questions and answers in an insightful and entertaining way.

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