J. Peter Campbell
There were several components to this system as diagrammed below.
Pasco motion sensor - measured z as a function of time.
Pasco photogate - measured the angular velocity as a function of time
Pasco force sensor - measured the force on the spring as a function of time. We did not use this much due to the fact that it is directly proportional to z (or -z, since the motion sensor was below the oscillating system).
The first step was to determine the physical properties of the system. Specifically, in order to model the trajectories of z and q as a function of time, we need to determine: k, delta, m and I.
m - The mass of the system
To determine the mass of the system, we needed only to place the entire bob onto a balance. Since we would need the individual masses for the moment of inertia calculation, however, we took it apart and massed each component separately. Though in introductory physics, one often neglects the mass of the spring in these calculations, in a real situation part of the spring's mass affects the oscillatory behavior. According to Berg and Marshall, approximating the contribution of the mass as 1/3 the mass of the spring is adequate. Using that estimation, we calculated the mass of the system to be m = 266g or .266 kg.
k - The longitudinal spring constant
Since we are assuming a linear restoring force, k can be calculated from the force equation:
F = kz (11)
Therefore, we averaged the F/z (or mg/z, where g = 9.8m/s2) ratio for six different masses hanging from the spring. We determined that k = 8.59 N/m.
I - Moment of Inertia
We estimated that the total I would equal the sum of the I's from the various components. Each shape has a unique moment of inertia formula, which we obtained from the Serway text. For this calculation also we had to contribute the moment of inertia from 1/3 of the mass of the spring. The moment of inertia also depends on the position of the small threaded beads on the perpendicular arm of the pendulum. This would allow us to vary the moment of inertia to determine the point of resonance later. By moving these beads, we found that we could vary the moment of inertia from Imin = 4.90E-5 kg*m^2 to Imax = 5.156E-5 kg*m^2.
delta - the rotational spring constant
This calculation was more complicated. We could not figure out a method of determine delta directly, so we had to estimate it from the definition:
wq2 = delta/I (12)
So, we measured the natural rotational frequency, wq, and solved for delta using our calculated moment of inertia, I. In order to find the rotational frequency, we needed to isolate the rotational motion from the naturally-coupled vertical motion. So, we connected the top and bottom of the spring with a string, which prevented vertical oscillations but allowed the spring to rotate. We found that the natural rotational frequency was 5.72 rad/s, which yielded a value of delta = .00163 N/m.
Once we had determined these physical characteristics, we were able to begin to study the dynamics of the system. We started by studying the trajectories of z(t) and wq'(t) as a function of I. We studied the relationship of the transfer of energy between the z and q by setting initial conditions where all of the energy at t=0 was in one coordinate. We chose z because it was easier. So, with the same initial conditions (z(0)=.10 m and q(0)=0 rad), we compared the plots of z(t) and q(t) in Data Studio.
We also experimentally measured the natural frequency of oscillation for the system with q(0)=0, which should be equal to w, from equations 8 and 9.
We measured the beat frequency for these plots to determine (using equation 10) the value for c.
Once we had determined c as a function of I, we were able to identify the experimental I that best satisfied the resonance condition, where wz = wq = w.
When we collected all of the experimental data necessary to find this condition, we attempted to model the motion of z(t) and q(t) using Mathematica. This system is exactly solvable for the initial conditions: w'(0)=0 & z'(0) = 0.
The solutions are:
where w1, w2 and w were determined experimentally. zo and qo are the initial conditions. The values for m and I were found above.
We also wanted practice modeling the system using the Mathematica NDSolve function, which we set up as below:
where k = spring constant, M = moment of inertia, d = delta, m = mass, e = coupling coefficient.