Pendulums

A wonder of the modern world

I was on the train from Slough to London pondering spin when I spotted a picture of a pendulum in a magazine.

I began to think that there are some similarities between spin and the behaviour of a pendulum and figured that it was definitely worth 20 minutes of my thinking time. Unfortunately the train to London was faster than my thought process and I arrived before I'd thought of anything interesting.

A couple of weeks later I am in a museum when I see an old antique grand father clock with a pendulum, so I gave it a little more thought, fortunately I had time to stop and think and this lead to this post.

The pendulum, discovered by Galileo apparently, is one of the most amazing inventions/discoveries ever. It completely changed the way we measured time.

You bung a weight on the end of a piece of string and you let it swing backwards and forwards. Provided that the swing angle is not too big then this thing will go on swinging in regular intervals for ages. Backwards and forwards, happy as Larry. This action is said to be isochronous, do like that word. It is this regular nature of the pendulum that allows its use in measuring time.

The equation of a pendulum is also rather straight forward, and is derived from Newton's second law of motion, F=ma, I got this from Wikipedia

L - length of the string
T - time period
g - local acceleration due to gravity
θ₀ - angle of the swing, small

What is amazing is that the actual period of the swing, the time it takes to swing from one side to the other and then back again, is directly proportional to the square root of the length of the string. Sorry, went into techno babble for a minute, what I meant to say was, the longer the string, the longer it takes to do one swing. The shorter the string, the shorter the time it takes to do a swing.

The time period of the pendulum is NOT dependant on the mass of the weight on the end of the string, 1 kg, 2 kg, provided it does not stretch the string, it does not matter what mass we use, the time period will stay the same. The only two factors we have to think about are the length of the string and gravity.

This is great, seriously, I genuinely can't help thinking there is something massively profound going on here, only we are not quite sure what that is yet.

Equation (1) above is very similar to another equation

C = 2π r  

which is just the equation for the circumference of a circle.

C - circumference of a circle,
r - radius of the circle

so if we take

r = √ L/g                                (2)

C = 2π √ L/g   

T =  2π √ L/g  

The circumference of the circle is the same as the time period, T. Which is cool in a way, as L gets smaller the circle will shrink and T will get smaller. L gets larger then the circle gets bigger and T increases.

This got me thinking about circles, area of circles, area of spheres and volume of a sphere, the equations for these are
area of circle, A =  π r² 

area of sphere, A_sphere =  4π r² 

volume of sphere, V =  4/3 π r³ 

using r in equation (2) above, we have


area of circle, A =  π L/g               (4)

area of sphere, A_sphere  =  4π L/g      (5)

volume of sphere, V =  4/3 π  √ L/g ³   (6)


If T from equation (1) is the time period, what are A, A_sphere  and V in equations (4), (5) and (6) physically? Do they actually represent anything at all? I am not sure at the moment. Will get the pencil and paper out and have a play. If I find anything interesting I'll put it in the next post.