The Physical Pendulum

The simple pendulum is a small mass on the end of a long string, length L, swinging with low amplitude in a plane. Then w = (g/L)^1/2 . The physical pendulum is the same except that the size of the object does not need to be small compared to L. It doesn't need a string at all; it could have a pivot on the object itself.

Call r_cm the distance from the pivot to the center of mass of the object. While swinging, if the center of mass is at an angle q from vertical, then torque r_cmmgsinq acts on it to make it accelerate toward equilibrium.
Apply t = Ia = Ia_cm/r_cm, and use x/r_cm for sinq . Recall that when a is proportional to x and they have opposite signs, simple harmonic motion (SHM) occurs, and
a = -w ²x. In this case, a = -r_cmmgx/I, so

w = (r_cmmg/I)^1/2. (equation 1)

Note that I is proportional to m, so w is independent of m. Note also that the above formula for w is valid only for very small oscillation limit, because only for this case the center of mass moves back and forth in a straight line. Note also that I of the simple pendulum is mL² and r_cm = L, so equation 1 reduces to w = (g/L)^1/2.

If you know I with respect to the center of mass (call this I_o), then I with respect to the pivot point (a distance r_cm from the center of mass) is I_o + mr_cm². This is the I to use in the equation for w .

Sometimes instead of a formula for I_o, you may be given a radius of gyration, r_g. This is the distance such that if all the mass were that far from the center of mass, the I_o would be the same. Thus I_o = mr_g².

While we are on the subject, another interesting distance is the center of percussion distance, a.k.a. the distance to the sweet spot, the spot on the bat where the batter in baseball wants to make contact with the ball. It turns out that the center of percussion is the same as the center of oscillation. The distance to the center of oscillation is the length of a simple pendulum which has the same frequency as the physical pendulum. Let's call the distance r_cp. It is easily shown that

I = mr_cmr_cp (equation 2) and

r_cp = r_g²/r_cm + r_cm (equation 3)

Now swing back to that sweet spot in cyberspace, my main page on oscillations and waves.

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mechanics
fluids, heat, electricity and magnetism
quantum
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Comments, questions: fredrick.gram @ tri-c.edu (remove spaces)