# Under what conditions is a pendulum a Simple Harmonic Oscillator, why?

For part of my lab write up on pendulum motion, my professor wanted us to find out why a pendulum was not a simple harmonic oscillator, and under what conditions it was. He also wanted to show this mathematically.

So far what I have is that if there is no damping(friction?) and if the the displacement is "small." I have no idea what "small" means. Can someone elaborate on this?

Keep in mind that this is my first semester in college and that this is my first calculus based physics class, so I don't know too much.

Right now we're on impulse and momentum. So anything beyond that I really don't know the math behind it, but maybe the idea yes. So if you do provide variables that are beyond that, please elaborate.

Thanks

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A simple harmonic oscillator is made by a ......(1) and a ......(2)
Write the force formula for these objects.

The pendulum is made by a ........ (1) and .....(3), something different from (2).
Write the force equation for 3.

Under which condition Force(2) ~= costant * Force(3) ?

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vela
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When you say some quantity is small, it's in comparison to some quantity which sets the scale of the problem. For instance, if you were talking about the collision between you and a flea, you could reasonably say the mass of the flea is negligible because it's small compared to your mass. Mathematically, you'd say mflea/myou<<1. On the other hand, if a problem involved you and the mass of the Earth, this time your mass would be negligible because myou/mEarth<<1. You should always be able to express smallness in terms of a dimensionless quantity being much less than 1.

Note the measure of an angle (in radians) is already dimensionless. It sounds kind of weird, but we really only write "radians" to keep track of the fact that some unitless number represents the measure of an angle.

The differential equation that governs a simple pendulum's motion isn't exactly the same as the one describing simple harmonic motion; however, with the right approximation, namely when sin θ≈θ, it reduces to the case of simple harmonic motion. So what you want to do is come up with a criterion for when sin θ≈θ is a good approximation.

A simple harmonic oscillator is made by a ......(1) and a ......(2)
Write the force formula for these objects.

The pendulum is made by a ........ (1) and .....(3), something different from (2).
Write the force equation for 3.

Under which condition Force(2) ~= costant * Force(3) ?
haha thanks. I did not think about it like that.

When you say some quantity is small, it's in comparison to some quantity which sets the scale of the problem. For instance, if you were talking about the collision between you and a flea, you could reasonably say the mass of the flea is negligible because it's small compared to your mass. Mathematically, you'd say mflea/myou<<1. On the other hand, if a problem involved you and the mass of the Earth, this time your mass would be negligible because myou/mEarth<<1. You should always be able to express smallness in terms of a dimensionless quantity being much less than 1.

Note the measure of an angle (in radians) is already dimensionless. It sounds kind of weird, but we really only write "radians" to keep track of the fact that some unitless number represents the measure of an angle.

The differential equation that governs a simple pendulum's motion isn't exactly the same as the one describing simple harmonic motion; however, with the right approximation, namely when sin θ≈θ, it reduces to the case of simple harmonic motion. So what you want to do is come up with a criterion for when sin θ≈θ is a good approximation.
Thanks