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Using Calculus to find Potential Energy

Using Calculus to find Potential Energy - U(0 = 0 We can...

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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: F s = - kx , where k is the spring constant. Let us also assign an arbitrary value to the potential at the equilibrium point:
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Unformatted text preview: 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 ) - 0 = - (- kx ) dx Implying that U ( x ) = kx 2 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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