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--------------------- 2400 - MATH - Equations are Sentences in Short Hand
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- Start with a simple equation for frequency and wavelength:
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------------------- f * w = CONSTANT
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------------------ This sentence says that the product of the frequency of oscillation, “f”, and the wavelength of the oscillating wave, “w”, is always equal to a CONSTANT for light and all other electromagnetic radiation. From Gamma Rays from distant stars to the alternating current that powers your house this equation holds true.
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- In order for the product to remain a CONSTANT number, if the frequency goes up the wavelength must go down. Frequency is cycles per second, if that increases, the wavelength of one cycle gets shorter in meters. Gamma Rays have very high frequency and very short wavelength. Radio waves have very low frequencies and very long wavelengths.
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-------------------- f * w = CONSTANT
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------------------- f * w = 300,000,000
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- In this case the CONSTANT is the speed of light, 300,000,000 meters per second. Units are left off for easier reading but all are in kilograms, meters and seconds.
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- Think of a product of two variables as equal to a CONSTANT area. Length times width equals a CONSTANT area. It the length gets longer the width must get shorter in order to keep the area constant. In Calculus math this concept is used in Integration.
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- Another simple equation for frequency and energy is:
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--------------- E / f = CONSTANT
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----------------- E / f = 6.6 * 10^-34
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- In order to keep this ratio remaining CONSTANT, if the frequency of “light” increases the energy also must increase. The constant is Planck’s Constant of Action. It is a very small number, (10^-34), so the frequencies must be very high in order to have a lot of energy in each photon.
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- “Light” is really meant to represent all electromagnetic energy. And the photon is the energy package or quantum of energy in each bundle of “light“. Light energy comes in quantums it is not continuous energy.
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- Gamma Rays and X-Rays have very high frequencies therefore their photons carry a lot more energy. Radio waves have very low frequencies, very long wavelengths, and radios need a lot of amplification in order to get enough energy to drive the radio’s speaker.
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- Think of ratios of two variables as the rate of change . Like velocity is the rate of change of distance per unit time. The ratio is a slope of rise over run if one of the variables is plotted versus the other variable. Watts per degree is the rate of change of watts of energy per degree of temperature. In Calculus math this concept is used in Differentiation.
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- Now let’s try an equation that is a little more complicated and takes several sentences to describe. The equation for the force of gravity:
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---------------------- F * d^2 = CONSTANT
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- The gravitational force between two objects of mass “ m” and mass “ M” , in this case the Earth and the Sun, is equal to the ratio of the product of the two masses and the Gravitational Constant to the square of the distance between their centers.
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- The force of gravity between two objects is proportional to the product of the masses of the two objects and inversely proportional to the square of the distance between them. Any time variables are proportional they can be turned into an equality by selecting the proper “constant of proportionality“. In this case the constant of proportionality is the Gravitational Constant, “G”.
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---------------------- G = 6.67 *10^-11
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---------------------- m = 6 * 10^24 mass of the Earth in kilograms
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---------------------- M = 2 * 10^30 mass of the Sun
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---------------------- CONSTANT = 79 * 10^43
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----------------- F * d^2 = 79 * 10^43
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- In order to keep the product of gravity force and distance squared constant, if one goes up the other goes down. The Earth orbit’s the Sun in an ellipse, not quite a perfect circle. So, the distance between the Earth and the Sun does vary somewhat. If the distance gets shorter the force gets larger.
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- What is keeping the Earth from spiraling into the Sun? Fortunately there is another law in physics called the law of Conservation of Angular Momentum. So, if the Earth gets a little closer to the Sun it speeds up in its orbit to increase its momentum and if the Earth gets farther away from the Sun is slows down to lessen its momentum. That is what keeps it in perfect orbit.
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- You can see from these equations where two variables are equal to a CONSTANT if you know the value of one variable you can always calculate the value of the other. If you know the distance to the Sun you can easily calculate the force of gravity pulling on the Earth. To calculate the values for other masses the full equation is:
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---------------------- F = G * m * M / d^2
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- One more equation then we will quit.
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----------------- p^2 / r^3 = CONSTANT
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- This equation says the period of an orbiting body squared is proportional to the radius of the orbit cubed. Remember if two variables have functions that are proportional they can be turned into equalities with the proper CONSTANT.
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- In this case for the Sun and the Earth. The period of a body in orbit is the time it takes to complete one revolution, one complete cycle. The ratio of the square of the period to the cube of the radius is always CONSTANT.
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-------------- p^2 / r^3 = 3*10^19
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- In order to keep this ratio of period and radius constant if one increases the other must increase. The Earth is 93 million miles from the Sun so the radius “r” equals 150 million kilometers.
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---------------- p^2 = ( 1.5 * 10^11) ^3 * 3*10^19
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---------------- p^2 = (3.35* 10^33) * 3 * 10^19
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---------------- p^2 = 10 * 10^14 seconds^2
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--------------- p = 3.16 * 10 ^7 seconds.
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-------------- p = 1 year
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- There are 31,600,000 seconds in a year so it takes one year for the Earth to complete one orbit.
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- If you want to learn more there are many more equations that can tell you a story. Let me know if you are interested and I can send you the math language reviews for these other equations:
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- The velocity needed to escape the gravitational pull of the Earth depending on the distance from the center. The further away you are the less velocity you need to have to escape the Earth’s gravity.
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------------------ v^2 * d = CONSTANT
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------------------ v^2 * d = 800,000,000,000,000
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- The ratio of the luminosity to the distance and temperature of a star. If you know the temperature and the distance to a star you can calculate its luminosity. Luminosity is the absolute brightness of a star, the total radiated energy per second.
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----------------- L / d^2 * T^4 = CONSTANT
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----------------- L / d^2 * T^4 = 0.0000000713
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- The product of the electric or magnetic force and the square of the distance between the charges is equal to a constant:
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-------------- F * d^2 = CONSTANT
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- The ratio of Energy to mass is a constant:
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-------------- E / m = CONSTANT
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- The ratio of Kinetic Energy to the square of velocity is constant:
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------------------- E / v^2 = CONSTANT
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- There are many equations. But they need not be so mysterious. Once reduced to the simplest sentences they usually make intuitive sense. But, not always. What is amazing about math is that when the equations to not match your intuition your intuition is usually wrong. Have fun with math.
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---------------- Footnotes:
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(1) The Constant speed of light is 299,800,000 meters per second. This review uses units of kilograms, meters, and seconds so the units are left off to make the reading a little easier.
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(2) Planck’s Constant of Action is equal to 6.625 * 10^-34 kilogram * meter^2 / second.
In different units it is also equal to 4.136 * 10^-15 electron volts * seconds / cycle.
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(3) The Gravitational Constant is “G” = 6.674215 *10^-11meters^3 / ( kg * sec^2)
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(4) p^2 = 4*pi^2 * r^3 / G * (m+M) = 39.5 * r^3 / 6.67*10^-11 * ( 2 * 10^30) = 3*10^-19
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(5) Science can not explain why the natural CONSTANTS are what they are. They are not derived mathematically, but, measured in the labs. They are weird numbers, or , quantities, but they are want the are to get the units to cancel and to get right variables in the right units. The other weird thing is that if the natural constants were just slightly different , a little more, or a little less, than life as we know it would not be possible.
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- June 11, 2019. 1235
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--------------------- Tuesday, June 11, 2019 -------------------------
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