Sorry to disagree. A swing has a natural frequency given by the length of the rope/chain/whatever. As you push it harder, it swings higher but completes a cycle in exactly the same time, as it accelerates for longer and achieves a higher maximum speed.

If you are interested in the period of pendulums the formula is Period in seconds = 2 x pi x square root(length of rope in metres/gravity or 9.81).

Of course this assumes a simplicity and perfection that doesn't exist in real life systems but its close enough to true to be observable in your local park. Take a stopwatch and a small kid and give it a go. :)

Linda, the formula you cite for the period of a pendulum T=2pi root L/g is a first approximation assuming the bob is a point mass, and the amplitude is small enough that sin theta = theta, and that the chain/istring has no mass. The equation applies only to a free swinging pendulum - one without damping or impulsing.,

Any of the standard applied maths text books will tell you that.. You will probably need a book on horology to find the rest of the story.
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To observe the effect you need two identical pendulums with heavy bobs (to reduce the effect of damping) started simultaneously but at different amplitudes.

You cannot illustrate this effect with a stopwatch in the lab or in the park becaue damping quickly reduces the amplitude to the point where "circular error" is undetectable...

The relevance to this thread is that a pendulum is not a resonator.because the period of the swing varies with the amplitude of swing...

I see no one has given you an answer on relationships beween natural frequency of vibration and phiysical dimensions. Linda.
There are relationships . They are deceptively simple. The algebraic notation needed to express these relationships is too complicated to be carried by this sort of email but let me have a go at the most simple.. For a string of length L vibrating laterally with both ends fixed, under a tension T with a unifrom load w per unit area including own weight, the natural frequencies are equal to

Kn/2pi root Tg/wl^2 where Kn = pi 2pi 3pi etc.

You will find these formulae tabulated in Roark and Young, Formulas for Stress and Strain ISBN 0-07--053031-9.

Other than in simple cases, these frequencies will be determined now a days by modelling using a finite element package. Try a serch for LISA.

Alll of which is exactly as I told Linda it would be. You cannot practically observe circular error with a single pendulum.. The amplitude decays too quickly. But, with two pendulums swinging simultaneously you can see the relative change of phase quite clearly

Horologists have been trying to make the pendulum isochronous since 1658
Only one man achieved it.

None of that has much to do with the original remark that a pendulum does not have a natural frequency of the sort asssociated with a resonator.

Last night I ran your experimental results together and plotted them. I then drew a least squares quadratic through them. The change in period with amplitude showed up straight away. I then drew the theoretical derived relationship on the same axes. Surprisingly the theoretical curve lay directly over the experimental results. The expression I used was , in the usual notation T=To (1+A^2/16) . This is the usual approximaton used instead of the full series expansion.

The point of all this has nothing to do with calculating the period of oscillation of the playground swing.

It is to do with the pendulum in the form of a garden swing not having a natural frequency and therefore not responding to constant frequency excitation. .the way that a resonator does.

Incidentally ,Christiaan Huygens is generally credited as the first person to try to make the pendulum isochronous. He used cycloidal chops. Bush Jackson and Philips came close. Fedosi Michaelovitch Fedchenko used a duplex suspension to get spectacularly successful results in 1952.

Aha - swing is a function of amplitude. That's why it's not cool to play too loud!

It don't mean a thing if it ain't got no swing! (with apologies to Duke Ellington!)

Wait, you fitted a quadratic to 4 data points, got a good fit and that is enough for you? There are no error bars. There is insufficient data. No physical justification for the particular fitting function. Where is the physical justification for chosing a quadratic? Did you simply choose a quadratic because it gave you the best fit?

Polynomials are frequently used for interpolants because of their ability to 'lay directly over the experimental results'. But this does not imply the existance of a relationship.

Incidentally, only one of the curves 'lay directly over the experimental results'.

Just FYI, this is NOT how physics is done.

No Sean. That is certainly not good enough for me and it is not what I did..

.I do have some idea of how physics is done - including the reduction of test results...

An important point here is that I was not trying to determine a law for the simple pendulum. That has already been done.. I was trying to use all of John Rayner's twenty four observations in a better way to see if his experiment in the park conformed with the existing widely accepted theory..

I had a closer look at things last night.
Two of the four data points in each data set are very close in amplitude and time. As the amplitude increases the damping is more pronounced and so the time values begin to diverge. Essentially there are only three unique points, as far as the fitting goes (in particular for the first data set).

Now, you can fit a quadratic very nicely through any three points (in the same way that a straight line will go perfectly through two points).

I again wish to point out that your analysis achieves nothing in the way of demonstrating any dependence of period on amplitude. As such there is no way to say whether or not the 'widely accepted theory' is upheld or not.
However, given that the 'widely accepted theory' is based on rigorus mathematical analysis of mechanical systems it stands on firm ground. Any criticim of this theory would necessarily require an equivalent level of analysis.

3 or 4 data points and a fit does not suffice.

Sean,
I was not trying to use John's data to demonstrate any dependence of of period on amplitude, or to say whether the theory is upheld or not..

If I must spell it out, the tone of John's contribution caused me to check whether John may have fudged his results..

An experiment to measure the period of a pendulum with a heavy chain consisting of rigid elements and with a rigid bob, etc produced results that were in aggregate so close to the theory for a simple pendulum.?. And no cofg calculations to be seen.

Incidentally, the period of a pendulum is increased only minutely by increasing the damping.. The principal effect of damping is to reduce the amplitude. and the period then decreases correspopndingly .

John,
Well. yes,light hearted and perhaps a little disparaging, that is the way I took it at the time, and in the light hearted spirit it would have raised a a chuckle if you had been pulling our collective leg.

Accorded attention ? What the heck. A bit of banter is fine.

Perhaps I could ask you to confirm that the observed periods are the average of 10 swings for the smaller amplitudes and of two swings for the larger amplitudes.

Alan

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