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Title:
10x-01 Physics in Action
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Description:
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When your classmates made a post in the forum
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that was a calder action. He wants to see
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examples of physics in real life, specifically
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simple harmonic motion. So I came to the park
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knowing that since simple harmonic motion is
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everywhere I find some example here and here
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I am in a tree. Turns out that when you displace a
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tree branch just slightly from equilibrium and
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release it, the resulting motion is simple harmonic
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motion. Don’t believe me, I can prove it to you.
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So, we’ve talked about simple harmonic motion,
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we’ve talked about masses on springs and we’ve
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talked about pendulums. Both of these when
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displaced from their equilibrium will exhibit simple
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harmonic motion and if we think that to why they
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display simple harmonic motion, we remember
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that it has something to do with some restoring
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force being proportional to a displacement. So for
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example for the mass on the spring, the restoring
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force was equal to minus K times X. The K was
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just a spring constant, X was the displacement
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from equilibrium and the minus sign, well the
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minus sign was very essential. The minus sign
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told us that the force was always opposite the
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displacement. So it tends to restore the mass to
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its equilibrium. Now this is thinking in terms of
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force. What about potential energy. Well for a
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spring the potential energy was equal to one half
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K times the displacement square and it’s this
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term the displacement squared that I want to talk
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about because we see that if we plot this,
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potential energy versus displacement we get this
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lovely parabola. Anything that has a parabolic
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potential energy curve when plotted against some
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sort of displacement will exhibit simple harmonic
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motion. When it’s displaced away from this
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equilibrium point, so if we can somehow show
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that a branch fluttering back and forth somehow
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exhibits this potential energy curve, well we’re
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done. We’ve proven that, it must be simple
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harmonic motion. Let’s see if we can do that.
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Well let’s think, what could the potential energy
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versus displacement look like for a branch and
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here when I say displacement, let’s say positive X
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means that the branch has been lifted up a little
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bit and negative means it’s been pulled down a
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little. To tell you the truth, I have no idea what this
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curve looks like. I know that it’s hard to bend a
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branch, so potential energy must somehow go up
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as I increase displacement, in fact in either
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direction. But then what does it do. Maybe there
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is some sort of plateau in the energy curve, the
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interpretation here would be, once we reach a
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certain displacement it’s not any harder to
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continue displacing the branch, to continue
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pulling it further in further out. I don’t think this is
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the case. Real branches don’t behave like that.
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Maybe instead it actually gets really, really
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difficult to continue bending the branch, or maybe
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it’s somewhere in between, of course these
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should be mirrored on this side. The fact is, we
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just don’t know. The only way we could figure this
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out, since branches are so complicated is by
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doing an experiment. But I am going to make the
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claim that we don’t need to because for small
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displacements, look what we have here and they
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can be proven mathematically in a very rigorous
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way that for small displacements this trough must
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has to be a parabola. So for this region, in here,
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potential energy is equal to something times
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displacement squared. And hell who really care
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what that something is and in fact what we’ve
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shown here is actually a deep truth of reality.
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Anything with some equilibrium position whether
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it’s a branch or a ball in a well or a mass on a
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spring for small displacements will with absolute
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certainty undergo a simple harmonic motion.
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So on oscillating tree branch, that’s my
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example of physics in action, what’s your's.