What is the expression for the period of a physical pendulum without the $\sin\theta\approx\theta$ approximation? i.e. a pendulum described by this equation:
$$
mgd\sin(\theta)=-I\ddot\theta
$$
Motivation for my question:
I'm asking this because I had an assignment in which I had to measure the period of oscillation of a physical pendulum and we always had to release it precisely from the same angle. So I'm wondering whether that precaution is irrelevant because it feels to me that the measurement error and error created by ignoring air drag is more significant than those that would emerge by not caring about being precise about the initial angle, as long as it is smaller than circa 30°. Am I right or wrong?
Best Answer
Sometimes, a good figure is worth more than a thousand equations :)
I numerically integrated the following equation of motion for a physical pendulum:
$$ I\ddot{\theta} + mgL\sin(\theta) + \frac12\mathrm{sgn}({\dot{\theta}})L\rho_{\mathrm{air}}C_DS(L\dot{\theta})^2 + \zeta\dot{\theta} + \gamma\theta = 0 $$
with $\mathrm{sgn()}$ the signum function. The second term in this equation is the torque exerted by gravity, the third term is due to air drag (which is assumed here to only act on the bob), the fourth term is due to friction at the attachment point, and the fifth term is a linear drag effect caused by simple bending of the string (therefore, I'm assuming the pendulum was constructed with a string).
I determined the progression of the pendulum's period simply by differencing the zero passes, times 2. I used the following values in the computations (which I think are pretty reasonable):
I normalized the periods thus determined, by dividing them by the period that follows from linearized theory for a compound pendulum (see the wiki, $T=2\pi\sqrt{I/mgL}$), and plotted the results for three cases:
for each case, I used three initial starting angles:
Here are the results:
So, in conclusion:
So I'd say you're right -- although the initial angle matters (which is what I think the exercise was intended to teach you), it doesn't matter as much as neglecting air drag does. It only starts mattering when you repeat the experiment in a vacuum chamber.