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

[MetalManuel]

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

 

[Quinzio]

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) ?

 

[vela]

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 m_flea/m_you<<1. On the other hand, if a problem involved you and the mass of the Earth, this time your mass would be negligible because m_you/m_Earth<<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.

 

[MetalManuel]

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 m_flea/m_you<<1. On the other hand, if a problem involved you and the mass of the Earth, this time your mass would be negligible because m_you/m_Earth<<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

 

[rwisz]

A pendulum is a simple harmonic oscillator when it follows the conditions of being an idealized system, meaning there is no external force acting on it and there is no energy loss due to friction or other factors. This means that the pendulum will continue to oscillate in a regular pattern indefinitely. In order for a pendulum to be a simple harmonic oscillator, the following conditions must be met:

1. Small amplitude of motion: The displacement of the pendulum from its resting position should be small. This means that the angle of swing should not be too large, typically less than 15 degrees. This condition is important because it ensures that the motion of the pendulum is linear and the restoring force is directly proportional to the displacement. If the amplitude is too large, the motion becomes non-linear and the pendulum will not follow the simple harmonic motion.

2. No damping: Damping refers to the loss of energy due to friction or other factors. In order for a pendulum to be a simple harmonic oscillator, there should be no damping present. This means that the pendulum should be in a vacuum or a medium with minimal resistance, allowing it to oscillate without losing energy.

3. Constant acceleration due to gravity: The acceleration due to gravity should remain constant throughout the motion of the pendulum. This ensures that the restoring force is always the same, resulting in a regular and periodic motion.

Mathematically, the equation for a simple harmonic oscillator is given by:

a = -ω^2x

Where a is the acceleration, ω is the angular frequency, and x is the displacement from the resting position. This equation shows that the acceleration is directly proportional to the displacement, and the negative sign indicates that the acceleration is always directed towards the equilibrium position. This is the condition for simple harmonic motion.

In conclusion, a pendulum is a simple harmonic oscillator when it follows the idealized conditions of small amplitude, no damping, and constant acceleration due to gravity. Any deviations from these conditions will result in non-linear or damped motion, making the pendulum not a simple harmonic oscillator.