**Wilberforce Pendulum**

**Mike Malenbaum**

**J. Peter Campbell**

__Setup__

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).

__Procedure__

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/s^{2}) 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 I_{min}
= 4.90E^{-5} kg*m^2 to I_{max }=
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:

w_{q}^{2}
=
delta/I
(12)

So, we measured the natural rotational frequency, w_{q},
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 w_{q}'(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 w_{z}
= w_{q}
= 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. z_{o} and q_{o}
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.