Background
A physical phenomenon described very simply with differential equations is the mass-spring system, where a mass is attached to a spring and allowed to oscillate. In an ideal system, the spring provides a restoring force proportional to its displacement from the rest position, or more commonly, F = kx. The equation that describes the motion of the mass where its position from rest, y(t), is:
m is the mass and k is the spring constant
The general solution of the above equation is a linear combination of sine and cosine terms, and the oscillations continue as t approaches infinity.
A more realistic system introduces the idea of damping, which slows the motion and dissipates energy from the system. A dampened harmonic oscillator can be modeled by the equation:
where b>0
Solutions to this equation are of the form
where C1 and C2 are constants.
The behavior of the solution is dependent on the value of the discriminant that appears in the exponent in the above equation. If the radical is an imaginary number, solutions contain sine and cosine terms, as well as exponential decay. If the radical is positive, the solutions are only exponential decay.
An interesting addition to the mass-spring system is the rubber band, which has a piecewise-defined restoring force, as follows:
This definition makes sense, seeing that a rubber band exerts a force when it is taut. But if the rubber band is completely slack, it exerts no force. Adding this term in, the overall mass-spring-rubber band oscillation equation is:
where
k1 is the spring constant and k2 is the rubber band constant
The lab recommends using a value of k1 between 12 and 13, which we took to be 12.5. k2, suggested to be between 4.5 and 5, was taken as 4.75.