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