# Simple harmonic oscillation

1. Sep 29, 2005

### daftjaxx1

Hi,
i'm trying to do this problem:

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A tympani drum has a billiard ball of mass m resting in the
middle. The billiard ball is displaced only vertically, very slightly
from its equilibrium, and will oscillate vertically around the
equilibrium position. The round rim of the drum is d in diameter, and
the drum-head is made tight by 8 turnbuckles that are each tightened
to a tension T, pulling the rim of the drum-head down around the
‘kettle’ body of the drum to tune it. What is the formula for the
restoring force? What is the oscillation frequency for the billiard
ball? Derive the equation of motion for this system. You may ignore the mass of the plastic sheet of the drum-head.

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my thoughts so far are as follows: this system is in simple harmonic motion which has an inertial force and a restoring force. let x be the distance the particle (in this case the billiard ball) is displaced. the inertial force is of the form

$$F_i = ma = m (d^2x/dt^2)$$ where a is the acceleration and m is the constant for inertial force

the restoring force is of the form $$F_r = -kx$$, where k is a constant measuring the "stiffness" of the oscillating material.

the oscillating frequency would be $$w^2 = \sqrt {k/m}$$

the equation of motion is simply
$$m(d^2x/dt^2) = -kx$$

so i know the general form of the equations for simple harmonic motion, but i'm not sure how to apply it to this specific problem. what is "m" and "k" in this case? i know that when the ball depresses the drum, the radial tension forces cancel but the downward component doesn't. the vertical motion should depend on the tension of the drum membrane and the angle formed. i just don't know where to go from here, how to put this all together. do i have to consider that the drum is held by 8 turnbuckles and calculate the force at each one somehow? i'm really confused. any help is much appreciated.

2. Sep 30, 2005