Swingin' around - a simple(?) physics question

Factual Questions
1

So I was sitting on a swing set the other day, recapturing my ancient childhood (alas, one tends to become cynical at the ripe old retirement age of 20), when I started thinking about the fact that it seemed to take longer to complete one full swing as I got progressively higher.

My vague memories of high school physics led me to the intuition that as you get higher, your velocity increases, but the distance you must travel increases as well, which should probably cancel out and leave you with the same period for each swing.

I stumbled upon a fancy-pants java-enabled page (http://hyperphysics.phy-astr.gsu.edu/hbase/pendl.html#c1) into which I plugged in some numbers and got the result that increasing the swing angle (I assumed the swing was more or less just a pendulum) does in fact lead you to a slightly longer period for the swing.

Can anyone confirm that that’s true and possibly explain why it’s the case that it works that way?

2

The period of a pendulum is:

T = 2pisqrt(l/g)

l = length of the string
g = gravitaional acceleration

It doesn’t depend on the mass or the displacement.

This eqation breaks down when the displacement is large compared to the length of the string.

3

right, from what I gather that equation depends on the simplification that sin(theta) ~ theta, but I am more interested in what things look like in the real world, where the displacement is large.

4

Well, it gets more complicated. The period is actually an infinte series:

2pisqrt(l/g)(1 + (sin^2(theta/2))(1/2^2) + (sin^4(theta/2))(1/2^2)(3^2/4^2) + …)

The short displacement period is just the first term. So for larger dispalecement, the period does depend on the displacement.

5

Much simpler explanation… I think it would have to do with some amount of pumping of the legs to increase or keep constant his velocity. I don’t care much for you city folks and your emme-vee squared over arr.

6

What you are remembering is the description of a simple harmonic oscillator. It only happens under a specific condition, namely when the restoring force is proportional to the displacement. A weight on a spring acts very much like this - if you pull the weight twice as far from the center (rest) position, the spring will pull it back twice as hard. In this case the period doesn’t depend on displacement.

A swing acts like a simple harmonic oscillator if the displacement is small. If A is the angle of the swing measured from vertical, the restoring force back towards the rest position is proportional to sin(A), and sin(A)=A for small values of A. However when displacement becomes large, restoring force increases more slowly (sin(A) < A for larger A). That is to say, if restoring force is 1 pound when the swing is at 1 degree from vertical it’s 9.95 pounds pounds at 10 degrees (very close to proportional) but only 49.6 pounds at 60 degrees (a full 20% lower than proportional). So the speed doesn’t increase very much, and period becomes longer.

7

Thanks! That’s exactly what I was struggling to understand.

Having just read “Surely You Must Be Joking, Mr. Feynman”, I’ve learned to be especially thankful for anyone who can explain scientific phenomena in simple, intuitive terms, and this definitely qualifies.

8

But, as per your post above, if you are interested in what happens when then displacement is large, it’s not a simple harmonic oscillator.

9

I like to think of extreme cases with problems like this. For a pendulum, the most extreme case is where the pendulum is completely upside-down (we’ll assume that the pendulum arm is rigid, not like a swing chain). Now, if this pendulum is balanced perfectly, it can, in principle, stay perched up there forever. And forever is, indeed, a rather long period.

So, we see that, to some degree at least, larger displacement means longer period.