Simple harmonic oscillation

[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.

 

[andrevdh]

Does this help a bit?
http://hyperphysics.phy-astr.gsu.edu/hbase/music/cirmem.html#c4

 

[quicknote]

Hi there,

First of all, great job on identifying the key components of simple harmonic motion - the inertial force and the restoring force. In this problem, the mass m refers to the mass of the billiard ball, and the constant k refers to the stiffness of the drum membrane.

To find the restoring force, we need to consider the forces acting on the billiard ball as it oscillates. As you mentioned, the vertical motion of the ball will depend on the tension of the drum membrane and the angle formed. We can use trigonometry to find the vertical component of the tension force at each turnbuckle, and then sum these forces to find the total restoring force.

To find the oscillation frequency, we can use the formula you provided, w^2 = \sqrt {k/m}. Since we are ignoring the mass of the drum membrane, we can use the mass of the billiard ball for m.

To derive the equation of motion, we can use Newton's second law, F = ma. In this case, the net force is the sum of the inertial force and the restoring force, so we have:

F = m(d^2x/dt^2) = -kx

Solving for x, we get the equation of motion:

x = A cos(wt + \phi)

where A is the amplitude of the oscillation and \phi is the phase angle.

I hope this helps you in solving the problem. Remember to carefully consider all the forces acting on the billiard ball and use trigonometry to find the vertical component of the tension force at each turnbuckle. Good luck!