# Kinematics – Variables Affecting Time Period in a Pendulum Other Than Length

experimental-physicsharmonic-oscillatorkinematics

I have been researching for almost a week looking for some variable that might affect the time period in a pendulum other than length and the only thing I found is the medium and that is due to resistivity. Does anyone have an idea of a variable that might have an effect on the time period of a pendulum?

Thanks a lot.

The initial condition affects the amplitude. From a free body diagram and using that the power is null (energy concervation $$\frac{\mathrm{d} E_{mec}}{\mathrm{d} t} = 0$$ ) you find as solution (forgetting the trivial solution $$\dot\theta = 0$$, where $$\theta$$ is the angle of the pendulum)

$$\ddot\theta = -\omega_0 \sin\theta ,$$

with $$\omega_0^2= \frac{g}{l}$$.

Then expanding the $$\sin$$ to first order (linear approximation) you'd find

$$\ddot\theta = \theta$$ which is an harmonic oscillator and has solution

$$\theta(t) = \theta_0\sin{\left(\omega_0 t\right)}$$.

With $$\theta_0$$ being the drop angle. Now you can already see that the period depends on the gravity and length (so you already found an other one, i.e. gravity).

But one can do more. The $$\sin$$ approximation is getting valid only when the angle gets small (how small?). One could get a better approximation by expanding to higher orders. $$\sin x = x - \frac{x^3}{6} + o(3)$$. Then one would look for solutions of the differential equation

$$\ddot \theta = \omega_0^2\left(\theta - \frac{1}{6}\theta^3\right)$$

sometimes called the Duffing equation. It requires advanced tools to be solved (such as Fourier series expansion). I can tell you that the solution would have a period

$$T = \frac{2\pi}{\omega_0}\left(1+\frac{\theta_0^2}{16}\right)$$.

So you see that with this better approximation the period of the pendulum depends on the drop off angle. Which intuitively makes sense.

If you don't want to solve the equations, you could also keep the $$\sin$$ formulation and solve the system numerically (implicit Euler or Runge-Kutta for instance) and you would see that the oscillations depend on the drop off angle. Or check by yourself with a pendulum (lab style).

If you have friction the energy will not be conserved and the period will not even be a constant of motion. The period would actually decrease in time.

I hope this helps :), Best