- 2738 - PHYSICS - Spacetime, Energy, and Calculus? Clocks run slower as you go faster and faster, yes, that includes biological clocks. NASCAR drivers are younger than their twin brothers the accountants . Using Einstein’s equations you can actually calculate how much younger. But, this review should be Newtonian, actually it is whole course in Physics.
---------------------- 2738 - PHYSICS - Spacetime, Energy, and Calculus?
- The Theory of Relativity teaches us that space and time are relative. Both are always relative to some reference plane. So, keep that in mind even though most of this review will be strictly Newtonian. In Einstein’s world space ,”s”, collapses and time,”t”, slows down as velocity approaches the speed of light, “c”. Velocity, c = s / t = space / time.
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- Velocity of light is meters / second, 300,000,000 meters/second. And, “c” is constant, regardless of the relative motion of objects. Therefore, if an object in space collapses like a pancake as it approaches the speed of light, “s” becomes smaller, and in order to keep “c” constant, time must get smaller too, fewer clicks of the clock.
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- Therefore, clocks run slower as you go faster and faster, yes, that includes biological clocks.
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- SPACE is measured as distance or length and the units are meters, symbol is “s“.
- TIME is measured as duration or elapsed time and the units are seconds, symbol is “t“.
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------------------------------- FOR OBJECTS WITH MASS:
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------------ VELOCITY is the rate of change of distance with time. v = d(s) / dt
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- Velocity is a change of space over a change of time. It is the derivative of space with respect to time. If you plot two variables distance versus time, the slope of the line on the graph is velocity, it is the rate of change of distance with respect to time. Calculus calls it a “derivative“, but you could just as easily call it a “rate of change” or a “slope” of one variable versus another. The units are meters / second.
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--------- ACCELERATION is the rate of change of velocity with time. a = d(v) / dt
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- Again, the little “d” refers to calculus and derivatives.
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- Acceleration is the second derivative of space with respect to time and the first derivative of velocity with respect to time. a = d^2(s) / dt^2
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- The units are meters / second / second, or, m / sec^2
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- The integral of acceleration is velocity. v = integral a (t)
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- INTEGRATION in CALCULUS is simply “ summation”. the summation of little areas. If you plot acceleration versus time, the y-axis is “a“, the x-axis is “t“. Then an area on the graph is “a * t” , a rectangle area is length times width. And, a = s/t^2, that is acceleration is meters/sec^2. And s/t^2 * t = s/t = velocity.
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- So, if you know acceleration and you integrate you get velocity. And, if you know velocity and you differentiate you get acceleration. That is almost all there is to CALCULUS, simple huh?
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--------------------------- ENERGY = mass * velocity^2
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--------------------------- E = m*c^2 = m * (9*10^16)
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--------------------------- The Energy of motion, KE = kinetic energy = ½ m* v^2
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--------------------------- Units of mass are kilograms
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--------------------------- Units of Energy are kg * m2 / sec^2
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----------------- Energy is mass times the (first derivative of space versus time) squared.
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- Energy can neither be created nor destroyed. Energy is Conserved. We have the same amount of Energy today as we did with the Big Bang.
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- With Relativity E = 1/(1-v^2/c^2)^½ * m * c^2. The new factor that is added, 1 / (1-v^2/c^2)^½ is called the Lorentz Factor. With this factor the amount of energy that is required to accelerate an object to the speed of light becomes infinite. Therefore, it can’t be done.
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--------------------------- Energy is also the capacity to do Work.
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--------------------------- ENERGY = E = m*v^2 = p*v = m * (ds/dt)^2 = A/t
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------------------------- WORK is a change in Energy, W = E1 - E2.
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--------------------------- Work is the product of a Force and distance, W = F*s
Units of Work are the same as units of Energy, kg*m^2/sec^2
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---------------------------- MOMENTUM is a product of mass and velocity, p = m * v
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--------------------------- Units for momentum are kilogram * meters/ sec
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--------------------------- Energy is momentum * velocity, E = p * v
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--------------------------- Units of Energy are kg* m^2 / sec^2
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- Energy is the momentum times the rate of change of distance with respect to time.
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- Remember, every time you see “ rate of change” “with respect to” you can replace that with “derivative“. E = p*ds/dt
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--------------------------- Momentum is the ratio of Energy / velocity. p = E/v
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- With Relativity momentum = p = 1/(1-v^2/c^2)^½ * m*v. An object can not be accelerated to the speed of light because its inertia goes to infinity as “v“ approaches “c“.
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---------------------------- FORCE is a product of mass and acceleration. F = ma
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--------------------------- Force = mass * d(v) / dt
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--------------------------- Force = mass * the rate of change of velocity with respect to time.
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- Force = mass * the 2nd derivative of distance with respect to time, F = m * d^2(s) / dt^2
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--------------------------- Force = kg * m / sec^2
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--------------------------- Energy is Force operating over a distance. E = F * s
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--------------------------- Force = Energy / distance. F = E / s.
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- Force = the rate of change of momentum with respect to time. F = d(p) / dt
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---------------------------- ACTION is a product of Energy and tim A = E*t
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--------------------------- Units of Action are = kg * m^2 / sec
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--------------------------- Energy = Action per unit period of time, kg*m^2/sec / sec
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--------------------------- Energy is the rate of change of Action over time, E = A /t
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----- Action is the product of Momentum and distance, (kg*m/sec )* meters A = p*s
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--------------------------- Action = Force * distance * time . Units are kg * m^2 / sec
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---------------------------- Action = mass * velocity^2 * time . Units are kg * m^2 / sec
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---------------------- Action = F*s*t = m*a*s*t = m*v^2*t = E * t = m*v*s = p * s
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- Acceleration is the derivative of velocity with respect to time. It is the slope of the line of velocity plotted versus time. INTEGRATION in CALCULUS is somewhat the opposite of derivatives. The integral of acceleration with respect to time is velocity, v = integral a*dt , a = d(v)/dt.
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- Distance is the integral of velocity with respect to time, s = integral v*dt, v = d(s)/dt
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- Action is the integral of momentum with respect to time,
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----------------------- Action = integral p*dt, p = d(A)/dt
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- WORK is the integral of force with distance, Work = integral F*ds. Work is equal to Force * distance, if the force is constant. But, if the force is varying over the distance you have to use calculus and take the integral of F*ds.
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-------------------------------------- FOR MASSLESS OBJECTS:
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------------------- PHOTONS have no mass, so E=mc^2 will not work. E=0.
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- But, the energy of a photon is not zero. We know photons have energy. We know photons have momentum. The classic demonstration I remember is the black and white vanes inside a clear light bulb. When a light is shined on the blades the black sides absorb the light and the white sides reflect the light and the net rotational force causes the blades to spin inside the light bulb.
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- We can still use the formula E=m*c^2 if we make some changes. m = E/c^2
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--------------------------- m = E/c*c
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--------------------------- m*c = E/c
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--------------------------- m*c is mass times velocity and that is momentum, so, p = E/c.
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- So, the energy of a photon is equal to its momentum times the speed of light.
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- We also know that the energy of a photon increases with its frequency. From radio waves to X-rays and Gamma rays the energy of photons increase with frequency. The breakthrough came in 1905 when Einstein and Planck determined that light energy was not continuous. It came in packets, bundles of energy, called quanta, and later called photons. Each photon had a fixed amount of energy.
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---------------------------- ENERGY operating over time is called “ Action”. A = E*t
And, each photon has a fixed amount of action. It is called Planck’s Constant of Action, and it has the symbol, “h”.
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------------------------------- h = 6.6 * 10^-34 kilogram*m^2 / sec
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------------------------------------------------- ACTION = A = E*t
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--------------------------- h = the action of one photon
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--------------------------- Energy of each photon depends on its frequency = E = h*f
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--------------------------- E = 6.6*10^-34 * f
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--------------------------- E = 4.136*19^-15 electron volts * f
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--------------------------- “c” = wavelength * frequency, meters / second.
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--------------------------- c= w*f
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--------------------------- E = h*c/w
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--------------------------- E = mc^2 = m*c*c = p*c, kg*m^2/sec^2
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------------------------- ENERGY of a photon = momentum * c, E = p*c
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--------------------------- E = m*c^2
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--------------------------- E = m*c^2 / (1-v^2/c^2)^½ as velocity approaches speed of light.
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---------------------------- Energy of a photon = h * f
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-------------------- Energy of a photon = 6.63 * 10^-34 * frequency, kg * m^2 / sec^2
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-------------- ENERGY = E = h*c / w = p*c / w = h*f = 6.6*10^-34 * f = 3*10^8 * p
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-------------- VELOCITY of electromagnetic radiation is a constant. The symbol is “c”
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------------- The velocity of a photon = wavelength * frequency. c = w * f = 3*10^8 m/s
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--------------------------- Units for a wavelength = meters / cycle
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--------------------------- Units for frequency = cycles / second
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--------------------------- Units for velocity is meters / second
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------------------- “c” = 299,792,458 meters / second, or, approximately = 3*10^8 m/sec
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- MOMENTUM of a photon = Planck’s Constant of Action / wavelength, p = h/w
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--------------------------- Momentum of a photon = 6.63 * 10^-34 kg * m / sec.
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The momentum of a light wave increases as the wavelength decreases, or momentum increases as the frequency of the light increases. (See note 2)
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--------------------------- p = h/w = h*f / c = E /c = E / 3*10^8 kg*m/sec
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- An example using these equations. Comparing the energy of yellow light to ultraviolet light, sunlight to sun tans.
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---------------------------- E = h*c/w w for yellow photons is 600 nanometers.
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------ Yellow E = 6.6*10^-34 * 3*10^8 / 6*10^-10 = 3.3 * 10^17 kg*m^2/sec^2
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----------------------------- Ultraviolet light photons have a wavelength of 400 nanometers.
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------ Ultraviolet E = 6.6*10^-34 * 3*10^8 / 4*10^-10 = 4.95 * 10^17 kg*m^2/sec^2
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- Therefore Ultraviolet light has 50% more energy than yellow light.
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- WAVELENGTH of a photon = w = c / f = h*c / h*f = h*c / E = h / p, meters
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- FREQUENCY = E/h = w /2*pi = 3*10^8/w , cycles per second
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- NUMBER of PHOTONS = Total Energy / h*f per photon, (See Review 2113)
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----------------------------- FOR QUANTUM SIZED OBJECTS:
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- The smallest unit of action = 6.6*10^-34 kg*m^2/sec
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- The smallest unit of mass = 1.1*10^-8 kg, m^2 = h*c/8*G*pi
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- The smallest unit of length = 1.6163* 10^-35 meters, s^2 = G*h/2*pi*c^3
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- The smallest unit of time = 1.4*10^-43 sec., t^2 = G*h/2*pi*c^5
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- The reason that there is a smallest unit of length is that the mass would turn into a black hole if the diameter, or length, got any smaller. And the reason that there is the smallest unit of time is that is the time it takes light to travel across that smallest unit of length. (See Review 2526)
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- (1) See Review 2562 - “Antimatter is all around us”
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- (2) Unit of Energy = joules = kg*m^2/sec^2
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------ Unit of Force = newtons = kg*m/sec^2
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- May 13, 2020 734 2738
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--------------------- Wednesday, November 11, 2020 -------------------------
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