# Question about a Pendulum's motion

• David Fosco
In summary: David, in summary, the statement that a pendulum will have the same period at different swing heights is an approximation used when the amplitude is small compared to the pendulum's length. Universities teach the more rigorous approach, which is only relevant for detailed analysis of actual pendulums. It's sad that teachers forget this meta-knowledge and teach something that is wrong to most people.
David Fosco
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

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"

David Fosco said:
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.

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
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
I commend Ibix's ambition, but estimate it is too much for the common non-scientist. For high school physics even the approximation is ambitious. University level at best.

## 1. What is a pendulum's motion?

A pendulum's motion is the back and forth movement of a weight or mass suspended from a fixed point by a string or rod.

## 2. What factors affect a pendulum's motion?

The factors that affect a pendulum's motion include the length of the string, the mass of the weight, and the angle at which the pendulum is released. Other factors such as air resistance and friction may also have an impact on the motion.

## 3. What is the relationship between the length of a pendulum and its period?

The length of a pendulum is directly proportional to its period, meaning that as the length of the pendulum increases, its period also increases. This relationship is known as the "period-length" relationship and was discovered by Galileo Galilei.

## 4. How does gravity affect a pendulum's motion?

Gravity is the force that pulls the pendulum weight towards the center of the Earth, causing it to swing back and forth. The strength of gravity can affect the speed and amplitude of a pendulum's motion.

## 5. What is the purpose of a pendulum in scientific experiments?

Pendulums are often used in scientific experiments to study the effects of gravity and other forces on a moving object. They are also used to demonstrate concepts such as periodic motion and conservation of energy.

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