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\nConservation of Mechanical Energy Proof (2DoF)\n

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\nHere is a quick math heavy proof of the conservation of mechanical energy in a 2 dimensional system. The proof in 3 dimensions is the exact same. Notice that the definition of the potential in the one dimensional system dV/dx = - F is...\nHere is a quick math heavy proof of the conservation of mechanical energy in a 2 dimensional system. The proof in 3 dimensions is the exact same. Notice that the definition of the potential in the one dimensional system dV/dx = - F is still consistent with the defubutuib for potential in the 2 dimensional system since Grad(V) in 1 dimension simply becomes dV/dx.

How does this all connect together with Work? Well, Work is defined by the integral of the force times infinitesimal displacement in the direction of the force. The total work is the work done by non conservative forces F(x,y) and non conservative forces. W = W_c + W_nc

W = integral F(x,y) . dr + W_nc

W = - integral grad(V) . dr + W_nc

W = - delta(V) + W_nc

Therefore work done by conservative forces is the negative change in potential energy.

But we're not done, we can prove W = W_c + W_nc = delta (T) where T = kinetic energy.

Therefore

delta(T) = - delta(V) + W_nc

delta(T) + delta(V) = W_nc

So the work done by non conservative forces is equal to the change in kinetic energy + change in potential energy.\n
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Resource Details

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Media Length
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6:29
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Grade
Higher Ed
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Subjects
Science
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\nVirtually Passed\n\n
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Date
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2019
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Language
English
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For Teacher Use
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Fine Print: Educational Use
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Concepts

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work, kinetic energy, potential energy, mechanical energy, force, speed, conservation, motion, vector
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conservative force, non conservative force, conservation of energy, pre multiply, double dot, dot del, bit messy, dot squared, integrational constant, squared, constant, equation, expression, term, function, conserved, particle, call, equal, purely, position, define, definition, write, likewise
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