Eric Murray, Spring 2006
Simple Harmonic Motion can be recognized in several ways. Perhaps most obvious is a sinusoidal dependance the position of an object on time
x = xmaxcos(ωt + φ)
where x is the position, xmax is the amplitude, ω is the angular frequency, t is time, and φ is the phase constant.
Finding the acceleration, a, by taking the second derivative of the position with respect to time yields
a = -ω2xmaxcos(ωt + φ) or a = -ω2x
Multiplying by the mass results in
ma = -mω2x so F = -kx
which is Hooke's Law, where k the spring or Hooke's Law constant, is mω2. So, if an object is subject to a linear restoring force, the object's position will vary sinusoidally with time. That is, an object on a spring is expected to undergo Simple Harmonic Motion.
With an arbitrary zero point of both force and position, Hooke's Law can be expressed as ΔF = -k Δx. Thus, k = -ΔF/Δx, or, k is the slope of a graph of F as a function of x.
The objects in this experiment will be known masses hanging from a vertical spring. You should read section 15.5 in your text to see the easiest way to deal with the effect of the gravitational force in such a situation. The spring's Hooke's Law constant will be determined three different ways, and the results compared.