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---------------------------- - 2265 - Deriving E = mc^2 with a teeter totter?
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- Every kid knows how a teeter-totter works. You lay a flat board on a fulcrum. If a boy and a girl weigh the same, put the fulcrum in the middle and the teeter-totter is perfectly balanced.
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- If one kid is heavier than the other simply move the board off center and put the heavier kid on the shorter side. The teeter-totter is still in balance.
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- If the kids both weigh the same but the boy had a heavy ball in his lap, then, he would have the short side to keep things balanced. If he threw the heavy ball to the girl on the left side she would go down unless we moved the fulcrum to the right to give her the short side.
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- This is the same simple concept we will use to derive E = m * c^2. Energy = mass * speed of light squared.
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- Our teeter-totter will need to have a weightless board and frictionless fulcrum. We will start with equal masses on each side of the teeter-totter then shoot a burst of light from one mass to the other.
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- The burst of light leaving one mass makes it lighter. The other mass absorbs the light and becomes heavier. The fulcrum is moved toward the second mass in order to restore the balance between the two masses.
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- Ok, let’s start the thought experiment. Are you thinking? You may need to sketch this on a piece of paper as we go through it. The two masses are balanced and the fulcrum is in the middle. We will call the fulcrum at zero distance. Then, the two masses are located at + x1 distance and – x2 distance.
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- The mass on the right of the teeter-totter gives a burst of radiation toward the mass on the left. The entire teeter-totter recoils in the opposite direction due to action equals reaction and shifts to the right.
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- When the radiation hits the left mass, the teeter-totter comes to rest, halting the movement to the right. The mass at the left has absorbed the radiation and becomes heavier. However, the fulcrum has shifted and the teeter-totter remains in balance. Both masses have shifted to the right by a distance Dx.
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- If the teeter-totter is in balance then the center of mass for the system has not moved. The center of mass is still at zero.
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- Before we started, the teeter-totter was in balance due to this equation:
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----------------------------- Mx1 + mx2 = 0
- If the radiation has mass then the mass at the right lost some amount of mass, Dm.
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- And, the mass at the left must have gained the same amount of mass, Dm.
- If the teeter-totter is still in balance then the new equation must still equal zero:
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--------------------- (m + Dm)( -x + Dx) + (m – Dm)(x + Dx) = 0
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-------------------- -2*Dm * x + 2* m* Dx = 0
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-------------------- Dm = m * Dx / x
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-------------------- Dx is the distance traveled.
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------------------- Dx = velocity * time (from distance = rate * time)
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- The velocity is the velocity of the radiation, and that is the constant speed of light = c.
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- The time is the time of the teeter-totter was moving to the right.
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- The distance across the teeter-totter is 2x, and the time it takes light to cross that distance is:
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------------------------ Time = 2x / c
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- The change in distance that shifted to the right:
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-------------------------- Dx = velocity * time = velocity * 2x/c
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- The change in mass due to the absorbed radiation, repeating the equation above:
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------------------------- Dm = m * Dx / x
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------------------------ Dm = m * velocity * 2 * x / x * c = 2m* velocity / c
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- The momentum of the entire teeter-totter is 2mv. The is from the law of conservation of momentum. The entire mass of the teeter-totter is 2m. The entire system must have the same momentum as when it moved to the right.
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---------------------------- Momentum = P = 2 * m * velocity
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--------------------------- Dm = P / c
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- The energy in a pulse of light is equal to momentum * speed of light:
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----------------------------------- E = P * c , and P = E / c
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---------------------------------- Dm = E / c^2
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- The change in mass, Dm, is the energy given off by the mass.
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- Therefore: E = m * c^2. Albert Einstein was right.
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- Energy is equal to mass times speed of light squared.
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- Energy is equal to mass times 90,000,000,000,000,000 meters^2 / seconds^2.
- That is a large number. No wonder the atomic bomb makes such a big hole in the ground.
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- February 10, 2019. 35
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--------------------- Sunday, February 10, 2019 -------------------------
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