# Experimental physics - simplified compound pendulum formula

## Homework Statement

We've been given an assignment to work out a formula for the period time of a compound pendulum. This is right in the beginning of an introductory physics course which means we haven't covered inertia or torque yet. So we're just supposed to find an approximate relation between the period and stuff we are "supposed to" know.

To help us, we have a big, assymetric metal plate with holes drilled in various places for pivoting, and some measuring instruments.

## The Attempt at a Solution

We did some measurements etc and found that the distance from the center of mass to the pivot point was to be our sole independent variable (for small angles, the amplitude had no impact).

When plotting the distance to the pivot and the period time, we found that it did not follow a traditional mathematical pendulum behaviour. The periodicity had a global minimum (at r = ~0.3m).

Some speculation and theorizing led us to the conclusion that as r tends to infinity, our function should tend to a mathematical pendulum, so we're probably looking for something like

$2pi\sqrt{r/g} + a/r^{b}$

Where a, b are positive real numbers.

We were now hinted to use log linearization to find b. Here's the problem though... I don't really understand how to do it when I have multiple terms on one side of the equation. Can anyone give me a clue?

rollingstein
Gold Member
Would Dimensional Analysis help?

I don't think it would, we're not trying to figure out which variables to use, just what exponent to raise it to and what constant to multiply it with.

rollingstein
Gold Member
I was questioning the form of your second term. How should it have units of time unless you want dimensional constants but that's messy?

rollingstein
Gold Member
Some speculation and theorizing led us to the conclusion that as r tends to infinity, our function should tend to a mathematical pendulum, so we're probably looking for something like

As r tends to ∞ doesn't your first term tend to ∞ too?

As r tends to ∞ doesn't your first term tend to ∞ too?

Yes, the first term tends to inf while the second term tends to zero, i.e. term one plus term two tends to term one.

I was questioning the form of your second term. How should it have units of time unless you want dimensional constants but that's messy?

We'll deal with that later.

:/ :/

rollingstein
Gold Member
$T=2pi\sqrt{r/g} + a/r^{b}$

$T - ( 2pi\sqrt{r/g} )= a/r^{b}$

$ln ( T - ( 2pi\sqrt{r/g} ) )= ln(a) - b* ln(r)$

Plot $ln ( T - ( 2pi\sqrt{r/g} ) )$ versus ln(r) and use the slope and intercept?

rollingstein
Gold Member
I'd just do a nonlinear numerical fit myself. Linearizing and plotting is a bit passe. :tongue2: