Conservation of Mechanical Energy Proof (2DoF)
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Here 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.
Here is a proof:
• Work Energy Proof Part...
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.
Пікірлер: 5
Can we use the same ideias in this video to extend the proof to 3 dimensions?
@abdulrahmann.9024
3 жыл бұрын
Yes, same algebra leads to same v but V(x,y,z)
@virtually_passed
3 жыл бұрын
Yes, the proof is the exact same since the vectors will just have a z component. The final answer will still be 1/2 m v^2 + V(x,y,z) = E (constant). Where v = sqrt(x.^2 +y.^2 +z.^2)
You miss an important point here. You assume that it is possible to construct V for arbitrary F(x, y) which is not true. If such a V exists, then F is a conservative force and the rest of the proof holds.
@virtually_passed
10 ай бұрын
You're correct! This proof assumes that such a potential exists