# Frequency of a simple pendulum

1. Sep 27, 2011

### bacon

A simple pendulum of length L and mass m is suspended in a car that is traveling with constant speed v around a circle of radius R. If the pendulum undergoes small oscillations in a radial direction about its equilibrium position, what will its frequency of oscillation be?

I don’t know how to post diagrams, so...
The negative x direction is to the left, toward the center of the circle of radius R. $\theta$ is the angle L makes with the vertical.
The forces on m are the gravitational force and the tension along length L. The forces result in SHM for small values of $\theta$.

$F_{x}=-kx=-mgsin\theta -m\frac{v^{2}}{R}sin\theta$

For small values of $\theta$, $\theta \approx sin\theta$.
$s=L\theta$. Where s is the arc length L sweeps through angle $\theta$.
For small values of $\theta$, $s \approx x$

Using these two approximations, the top equation becomes
$kx=mg\frac{x}{L} +m\frac{v^{2}x}{RL}$
After cancelling the x's and rearranging, I get
$\frac{k}{m}=\frac{g+\frac{v^{2}}{R}}{L}$
For frequency f and period T, $f=\frac{1}{T}$, $T=2 \pi \sqrt{\frac{m}{k}}$,
$f=\frac{1}{2 \pi}\sqrt{\frac{g+ \frac{v^{2}}{R}}{L}}$
Unfortunately, the answer in the back of the book is
$f=\frac{1}{2 \pi}\sqrt{\frac{\sqrt{g^{2}+\frac{v^{4}}{R^{2}}}}{L}}$

Thanks for any help.

Last edited: Sep 27, 2011
2. Sep 28, 2011

### ehild

The pendulum will oscillate around an angle θ0 because of the centrifugal force acting in the car as rotating frame of reference: tan(θ0)=v2/(Rg). The deviation from this angle can be considered small, but not the angle with respect to vertical.

ehild