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sophiecentaur

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The bob only has one degree of freedom - along the arc of the circle. So there is only one dimension involved. It is true, of course, that the restoring force is only proportional to the displacement for small angles and the motion is not purely sinusoidal in time

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olivermsun

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I posted a response in your previous thread. Just to reiterate: for a pendulum the "restoring force" is gravity itself, so the potential energy of the oscillator is just the gravitational potential energyIf we apply the small angle approximation so that a simple pendulum can be considered to be under going SHM, what kind of potential energy would the pendulum bob be having? ...

One way to formulate this problem is to pick as your single dimension θ(

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sophiecentaur

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If you look at what happens to a mass on a coil spring, there can also be rotation around the axis of the helix. This gives another dimension in that system too - if you want to find one.

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The bob only has one degree of freedom - along the arc of the circle. So there is only one dimension involved. It is true, of course, that the restoring force is only proportional to the displacement for small angles and the motion is not purely sinusoidal in time

But if the motion of the bob along the arc is one dimensional, then circular motion is one dimensional but it's clearly two dimensional. (I'm sorry for my misunderstanding regarding the difference between a dimension and a degree of freedom if there is any, I'm still in my senior year in high school, starting university next year.)

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olivermsun

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nasu

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Ok, then maybe is just a matter of a too narrow field of view, so to say.But if the motion of the bob along the arc is one dimensional, then circular motion is one dimensional but it's clearly two dimensional. (I'm sorry for my misunderstanding regarding the difference between a dimension and a degree of freedom if there is any, I'm still in my senior year in high school, starting university next year.)

You can have harmonic oscillations in any number of dimensions. Dimensionality is not really an issue. See for example the 3D harmonic oscillator.

The important feature of harmonic motion is the proportionality of restoring force to the displacement from equilibrium. No matter how you measure this displacement.

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Note the dictionary definition of extension - "a measurable extent of a particular kind". So θ is as much a dimension as x or y.

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Hmmm..so, acceleration would still be a = -ω

Note the dictionary definition of extension - "a measurable extent of a particular kind". So θ is as much a dimension as x or y.

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And the potential energy would be mgx? Shouldn't x be vertical then? I'm confused.

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How would we have gravitational potential energy if there were no vertical?

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sophiecentaur

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Have you been reading what's been written about the fact that the co ordinate system you use, is arbitrary? Of course the PE changes with angle but angle is the best variable to analyse the motion with. You need to learn to be flexible about these things and not try to prove to yourself that the text books have been getting it wrong. They haven't; you are just not looking at things in the best way to get a (valid) answer.How would we have gravitational potential energy if there were no vertical?

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