# Question about a Pendulum's motion

It was always my understanding that a Pendulum has equal time at different swing heights and teachers teach that but in fact, it is not true. I downloaded the LabinApp Pendulum Amplitude Demo App and it shows a slightly different time as you drop higher and higher. My question is why don't they teach this and for the people that understand the Math show as you increase the height this is the part that changes the outcome in time per swings. Thank you for your time.

berkeman

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In a physics curriculum the harmonic oscillator is a very important topic: lots of systems have a restoring 'force' that is proportional to the deviation from equilibrium. It is the first order analysis of any system with a minimum in the potential function. The convenient approximation that turns a pendulum into a harmonic oscillator is ##\sin x \approx x##, which is a good approximation as long as ##{x^3\over 3!}<<x##, so for quite a reasonable range.

The more rigorous approach you hint at is only relevant for detailed analysis of actual pendulums (pendula?), a much more restricted area.

Thank you for your reply... When I took science in high school I distinctly remember my teacher saying that it didn't matter how high you raised a pendulum, it would always have the same period. If you look at videos on YouTube you can see tons of teachers telling their classes this same thing. It seems as if this is supposed to common knowledge. And for very small angles it is. But they will show a pendulum raised to horizontal and proclaim that the period will be the same no matter how high. I'm just curious as to why teachers would teach something so wrong and be so sure about it? They don't even say "approximately" or "at smaller angles"

Mister T
Gold Member
My question is why don't they teach this
Every college-level introductory physics textbook will make it clear that the statement is an approximation used when the amplitude is small compared to the pendulum's length. Do a google search for brachistochrone, that's the shape, not the arc of a circle, that gives you the property described by your teacher. As long as the arc length is small compared to the radius, the two curves match up very closely.

It's quite possible your teacher didn't know this and didn't learn it when he had the chance.

Borek
Mentor
When the pendulum theory is taught at the basic level "for small amplitudes" is a fine print, and as all fine prints it gets forgotten and/or neglected till almost nobody remembers it ever existed. I agree it is sad.

anorlunda, vanhees71 and Ibix
Ibix
It's more than sad. Teaching people that there are simple results and there are complicated results, why we use the simple ones, and when we have to resort to the complicated ones, is one of the things I think a basic science education should teach. In my opinion, that kind of meta-knowledge about science and the process of science is far more important to most people than a harmonic oscillator, no matter how useful the thing is to quantum and thermodynamics.

End rant.

vanhees71
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