Simple Harmoinc Motion theory Question? (1 Viewer)
- Thread starter Currybear
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The short answer is a pendulum moves in simple harmonic motion only when the angle is small, and the spring and mass moves in SHM.
The more detailed answer involves talking about what exactly 'doing physics' means. In physics, we try and model reality via mathematical models. However reality is messy and so we inevitably have to make assumptions and simplifications, but we do this while still trying to keep our models close to what we observe experimentally. When we model a pendulum, the assumptions we usually make are that the string connecting to the bob is massless and inelastic, the pendulum is frictionless so it doesn't lose energy, etc... under these assumptions we can derive the equation
where k is a constant and theta is the angle the pendulum makes with the vertical.
Now the motion described by this equation is certainly harmonic, the further theta is away from 0, the stronger the force will be to push it back towards 0 (i,e. its 'rest' position). But its not simple harmonic. SHM occurs when the equation is of the form
, or in english, the restoring force is directly proportional to the its displacement from rest. Proportionality is the key difference between these two harmonic motions, this just means that if you double the distance from its rest position then the restoring force pushes back twice as hard. Three times the distance, three times as hard.
But if we make the further assumption that theta is small, then we can approximate \approx \theta$)
. You can see this if you graph y=x and y=sin(x), for small x the two graphs almost over lap. So for small theta, we can use the SMH equation to model the motion of the pendulum. Does this mean the pendulum moves in simple harmonic motion? no it doesn't. But when theta is small and our pendulum has properties that are close to our other assumptions, the equation will give us answers that will be close to what we observe experimentally.
In reality, nothing actually moves in simple harmonic motion. Not even the spring and mass. When we model the spring and mass we also make a bunch of assumptions, one of which is usually called hooke's law which basically says that
. So one of the assumption is that it moves SHM. This isn't a very good model in practice though (atleast when compared to the simple pendulum model), as most spring and mass systems have quite a bit of friction, so the mass will stop oscillating quite fast and is obviously not moving simple harmonically.
The more detailed answer involves talking about what exactly 'doing physics' means. In physics, we try and model reality via mathematical models. However reality is messy and so we inevitably have to make assumptions and simplifications, but we do this while still trying to keep our models close to what we observe experimentally. When we model a pendulum, the assumptions we usually make are that the string connecting to the bob is massless and inelastic, the pendulum is frictionless so it doesn't lose energy, etc... under these assumptions we can derive the equation
Now the motion described by this equation is certainly harmonic, the further theta is away from 0, the stronger the force will be to push it back towards 0 (i,e. its 'rest' position). But its not simple harmonic. SHM occurs when the equation is of the form
But if we make the further assumption that theta is small, then we can approximate
In reality, nothing actually moves in simple harmonic motion. Not even the spring and mass. When we model the spring and mass we also make a bunch of assumptions, one of which is usually called hooke's law which basically says that
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that explains a lot