Using newtons second law m we obtain the equation of

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Unformatted text preview: harmonic motion is shown in Figure 1. Using Newton’s second law, ￿ = m￿ , we obtain the equation of motion F a m a = −k x, 85 (1) Figure 1: These two graphs are defining features of simple harmonic motion. The force law is linear with respect to the displacement, x, and the potential energy function is a parabola. You can think of simple harmonic motion as a situation where a ball sitting at the bottom of the potential energy “well” is displaced slightly. What would happen? It would roll back and forth periodically, and its x-position would be a sine curve. where x is the object’s position and a is its acceleration. For those familiar with calculus, a is the second derivative of x with respect to time, so m d2x = −k x. dt 2 All systems that undergo simple harmonic oscillation have equations of motion of this form. Solutions to this equation have the form x(t ) = A sin(ω t + δ ), where δ is a constant that depends on the time we set to be t = 0 (this is arbitrary), and ￿ ω = k /m is the angular frequency of the object’s oscillation. The angular frequency of the oscillation is related to the frequency, f (the number of oscillation cycles the object undergoes each second) and the period, T (the amount of time it takes the object to undergo one complete oscillation cycle) via the relations f= ω 1 =. 2π T (2) The defining feature of simple harmonic motion is the force law F = −kx. The potential energy 86 graph associated with this force law is a parabola (see Figure 1), and the curvature of this parabola is defined by k; if an object moves a distance x away from its equilibrium position, it gains an amount of potential energy 1 U = kx2 . (3) 2 Note that the force the object experiences at a displacement x is the negative slope of the potential energy function at that point. Interestingly, these properties are very closely related to similar concepts from uniform circular motion. In fact, if one considers an object moving in uniform circular motion, its pr...
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