Simple Harmonic Motion Explained

If we setup an F = ma equation for a spring in simple harmonic motion but replace a with the second derivative of position, x, we get the following;

 -k d2x x = m dt2

Eq.(m9-5)

This equation, which relates x to its second derivative, is called a second level differential equation. If you have any knowledge of derivatives, or have read through the derivative section of our basics tutorial, you know that the second derivative of a sine (or cosine) function gives us a negative sine (or cosine). The only explanation for the -k/m is that the argument of the sine (or cosine) function must be not only time, but the product of ω and time because taking the derivative twice would bring the omega out as a coefficient of the sine (or cosine) twice and would therefore be ω2. Also, if we only insert a sin(ωt) (or cos(ωt)) for a, we get the maximum displacement to be 1. This is not necessarily true so we must add a constant coefficient to represent the maximum displacement, or amplitude, A. Because both sine or cosine will work for this second level differential equation, both must be included in a general equation for the position as a function of time. Therefore we end up with the equation

 x(t) = Asin(ωt) + Bcos(ωt)

Eq.(m9-6)

Because we know that velocity is the derivative of position, we can get an equation for velocity as a function of time by taking the derivative of the position equation.

 v(t) = Aωcos(ωt) – Bωsin(ωt)

Eq.(m9-7)

The proof for simple pendulums (in which there is simply a mass attached at the end of a massless string) is quite similar and yields almost identical results. One major difference is that the proof for simple pendulums, and all the equations for simple pendulums, only work at very small angles. This is because the equations and proof hinge upon the small angle theorem which states that, when working in radians, if the angle is very small than the angle divided by the sine of the angle is very close to 1. This is really saying that at small angles, the slope of the sine vs angle graph is 1. The equations for simple pendulums are identical to the equations for springs in simple harmonic motion except that k is replaced with g, the acceleration due to gravity, and m is replaced with l, the length of the string. Therefore we see that the oscillation of a pendulum is not dependent of the mass of the object attached to the end of the string.