**Wilberforce Pendulum**

**Mike Malenbaum**

**J. Peter Campbell**

__Theory__

To simplify the system, we assume that the motion is characterized by a linear restoring force in each direction, z and q. With the addition of a coupling factor, c, the equations of motion for the system become:

** **
**
(1,2)**

where m is the mass of the system, k is the spring constant for the z axis, I is the moment of inertia, delta is the rotational spring constant.

Therefore:

**
**(3,4)

The first step in solving this system is to assume oscillatory motion, so that the solutions for z(t) and q(t) are:

**
**
(5,6)

Plugging these solutions into equations 3 and 4, respectively, yields:

**
**(7)

Since the Wilberforce pendulum contains two degrees of freedom, the solution has two normal modes. Equation 7 can be solved to give the frequencies of the normal modes:

**
**(8) ** **
**
**(9)

w_{1}-w_{2} is defined as
w_{b}, the beat frequency,
which is the frequency of the transfer of energy between the two modes.

Further, equations 8 & 9 can be used to determine the coupling as a function of the beat frequency, mass, and moment of inertia:

**
** (10)

From this equation, we can calculate c for various I values to determine the I at which resonance occurs, or energy is most completely transferred between longitudinal and rotational motion.

Assuming sinusoidal oscillations, one can solve for z(t) and (t), given the
initial conditions, z_{o} and q_{o}.
This system is exactly solvable for the initial
conditions: w'(0)=0
& z'(0) = 0. The solutions are:

**
**(11)

**
**(12)

where w1, w2 and w were determined experimentally.

This can be further simplified to provide the relationship between z_{o} and q_{o}
for the normal modes w_{1} and w_{2} (equations 8 & 9).

**
** (13,14)

The constant is defined as , which is called the radius of gyration, simply the radius of the spring. Therefore, the normal modes of the pendulum are to a close approximation:

**
** (15,16)