Potential Energy of Springs

The force of a spring is determined by Hooke's Law, which is

Fsp=-kΔx,

where Fsp is the force of a spring, k is the spring constant (measured in newtons/meter), and Δx is the displacement of the spring. Moreover, the force of a spring is equivalent to

Fsp=-kx=-dUsp/dx.

In other words, the force of a spring is equivalent to the change in potential energy of the spring, with respect to distance [3]. If we were to solve this differential, we would get


Usp=1/2kΔx2

which means the potential energy of a spring is proportional to the displacement of the spring squared [3]. It was through the work of Calculus made people easily understand the relationship between potential energy and springs.

Simple Harmonic Motion
Earlier we showed that the force of a spring is equivalent to the spring constant times the displacement. In terms of Newton's Second Law, this means the force of a spring equals mass times acceleration, or rather,

ma=-kx,

where m is the mass of the spring and a is the acceleration of the spring (Δx=x, assuming the spring starts from equilibrium). This is assuming there are no other external forces acting on the spring (friction, for example, would make the spring move slower)! This happens to imply

a+(k/m)x=0 [2].

So what's the big deal? One might think that calculating the acceleration of a spring is very hard, so this doesn't really help. This may be true, except we happen to know the equation of the position of a spring with respect to time, which is

x(t) = A sin(ωt+ φ) ,

    where A is the amplitude of the spring, ω is the angular velocity of the spring, t is the time, and φ is the phase shift of the spring (the phase shift isn't that important right now). From this equation, we may find that the acceleration of a spring will be
a(t)=d2x/dt2=-ω2A sin(ωt+ φ).

Plugging this into a+(k/m)x=0, we get that 0=(k/m)A sin(ωt+ φ) -ω2A sin(ωt+ φ), which implies (k/m)1/2 [3].
In other words, the angular velocity of a spring is equivalent to the square root of the spring constant divided by the mass of the spring. Perhaps this is not the most exciting discovery in physics, but through some simple manipulation of physics equations, we can get a better understanding of how springs work.                                                                                                              
Oscillating
              Spring GraphSpring
            Oscillating
(The graph on the left is an example of an oscillating graph. As you can see the shape of the graph looks like a spring that is stretched out, but it's really a sine function. From this graph we should be able to determine the period, amplitude, and position of a spring based on its spring constant and mass. The animation on the right shows a spring oscillating. You may notice that the position of the spring varies with respect to time. In the sine function, the graph reaches a maximum and minimum height, just like the spring in the animation.)

    We will now discuss the conservation of momentum to continue our understanding of calculus in physics.



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