Conservation of Mechanical Energy
We have just established that ΔU = - W, and we know from the Work- Energy Theorem that ΔK = W. Relating the two equations, we see that ΔU = - ΔK and thus ΔU + ΔK = 0. Stated verbally, the sum of the change in kinetic and potential energy must always equal zero. By the associative property, we can also write that:
|Δ(U+K) = 0|
Thus the sum of U and K must be a constant. This constant, denoted by E, is defined as the total mechanical energy of a conservative system. We can now generate a mathematical expression for the conservation of mechanical energy:
|U + K = E|
This statement is true for all conservative systems, and thus for all systems in which U is defined.
With this equation we have completed our proof of the conservation of mechanical energy within conservative systems. The relation between U, K and E is elegantly simple, and is derived from our concepts of work, kinetic energy, and conservative forces. Such a relation is also a valuable tool in solving physical problems. Given an initial state in which we know both K and U, and asked to calculate one of these quantities in some final state, we simply equate the sums at each state: Uo + Ko = Uf + Kf. Such a relation further bypasses our kinematics laws, and makes calculations in conservative systems quite simple.
Using Calculus to find Potential Energy
Our calculation of the gravitational potential energy was quite easy. Such an easy calculation will not always be the case, and calculus can be a great help in generating an expression for the potential energy of a conservative system. Recall that work is defined in calculus as W = F(x)dx. Thus the change in potential is simply the negative of this integral.
To demonstrate how to calculate potential energy using vector calculus we shall do so for a mass-spring system. Consider a mass on a spring, at equilibrium at x = 0. Recall that the force exerted by the spring, which is a conservative force, is: Fs = - kx, where k is the spring constant. Let us also assign an arbitrary value to the potential at the equilibrium point: U(0) = 0. We can now use our relation between potential and work to find the potential of the system a distance x from the origin:
|U(x) = kx2|
This equation is true for all x. A calculation of the same form can be completed for any conservative system, and we thus have a universal method for calculating potential energy.
Though Newtonian mechanics provide an axiomatic basis for the study of mechanics, our concept of energy is more universal: energy applies not only to mechanics, but to electricity, waves, astrophysics, and even quantum mechanics. Energy pops up again and again in physics, and the conservation of energy remains one of the fundamental ideas of physics.
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