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- [Instructor] So, we saw that for a mass

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oscillating on a spring,

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there's a certain amplitude

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and that's the maximum
displacement from equilibrium.

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But there's also a certain period,

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and that's the time it takes
for this process to reset.

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In other words, the time it takes

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for this mass to go
through an entire cycle.

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But what do these things depend on?

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We know the definitions of them,

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but what do they depend on?

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Well, for the amplitude,
it's kind of obvious,

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the person pulling the mass back.

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Whoever or whatever is
displacing this mass

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is the thing determining the amplitude.

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So if you pull the mass back far,

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you've given this oscillator
a large amplitude,

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and if you only pull it back a little bit,

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you've given it a small amplitude.

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But it's a little less obvious
in terms of the period.

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What does the period depend on?

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Who or what determines the period?

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Maybe it depends on the amplitude,

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so let's just check.

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If I asked you, if I asked you,

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if I pulled this back farther,

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if I increase the amplitude farther,

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will that change the
period of this motion?

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So, let's think about it.

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Some of you might say, yes,

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it should increase the period

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because look, now it has
farther to travel, right?

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Instead of just traveling
through this amount,

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whoa that looked horrible,

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instead of just traveling
through this amount

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back and forth,

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it's gotta travel through
this amount back and forth.

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Since it has farther to travel,

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the period should increase.

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But some of you might
also say, wait a minute.

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If we pull this mass farther,

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we know Hooke's law says that
the force is proportional,

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the force from the spring,

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proportional to the amount
that the spring is stretched.

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So, if I pulled this mass back farther,

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there's gonna be a larger force

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that's gonna cause this mass
to have a larger velocity

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when it gets to you,

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a larger speed when it gets
to the equilibrium position,

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so it's gonna be moving
faster than it would have.

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So, since it moves faster,

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maybe it takes less time for
this to go through a cycle.

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But it turns out those two
effects offset exactly.

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In other words, the fact that this mass

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has farther to travel

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and the fact that it will
now be traveling faster

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offset perfectly and it doesn't
affect the period at all.

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This is kinda crazy

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but something you need to remember.

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The amplitude, changes in the amplitude

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do not affect the period at all.

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So pull this mass back a little bit,

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just a little bit of an amplitude,

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it'll oscillate with a certain period,

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let's say, three seconds,

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just to make it not abstract.

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And let's say we pull
it back much farther.

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It should oscillate
still with three seconds.

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So it has farther to travel,

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but it's gonna be traveling faster

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and the amplitude does not affect

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the period for a mass
oscillating on a spring.

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This is kinda crazy,

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but it's true and it's
important to remember.

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This amplitude does not affect the period.

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In other words, if you were
to look at this on a graph,

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let's say you graphed this,
put this thing on a graph,

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if we increase the amplitude,

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what would happen to this graph?

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Well, it would just
stretch this way, right?

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We'd have a bigger amplitude,

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but you can do that and
there would not necessarily

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be any stretch this way.

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If you leave everything else the same

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and all you do is change the amplitude,

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the period would remain the same.

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The period this way would not change.

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So, changes in amplitude
do not affect the period.

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So, what does affect the period?

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I'd be like, alright, so the
amplitude doesn't affect it,

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what does affect the period?

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Well, let me just give
you the formula for it.

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So the formula for the
period of a mass on a spring

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is the period here is gonna be equal to,

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this is for the period
of a mass on a spring,

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turns out it's equal to two pi

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times the square root of the mass

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that's connected to the spring

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divided by the spring constant.

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That is the same spring constant

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that you have in Hooke's law,

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so it's that spring constant there.

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It's also the one you
see in the energy formula

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for a spring, same spring
constant all the way.

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This is the formula for the
period of a mass on a spring.

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Now, I'm not gonna derive this

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because the derivations
typically involve calculus.

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If you know some calculus

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and you want to see how this is derived,

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check out the videos we've
got on simple harmonic motion

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with calculus, using calculus,

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and you can see how this
equation comes about.

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It's pretty cool.

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But for now, I'm just gonna quote it,

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and we're gonna sort of just
take a tour of this equation.

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So, the two pi, that's
just a constant out front,

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and then you've got mass here

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and that should make sense.

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Why?

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Why does increasing the
mass increase the period?

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Look it, that's what this says.

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If we increase the mass, we
would increase the period

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because we'd have a larger
numerator over here.

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That makes sense 'cause a larger mass

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means that this thing
has more inertia, right.

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Increase the mass, this
mass is gonna be more

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sluggish to movement, more
difficult to whip around.

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If it's a small mass, you can
whip it around really easily.

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If it's a large mass,
very mass if it's gonna be

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difficult to change its
direction over and over,

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so it's gonna be harder
to move because of that

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and it's gonna take longer to
go through an entire cycle.

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This spring is gonna
find it more difficult

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to pull this mass and then slow it down

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and then speed it back up
because it's more massive,

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it's got more inertia.

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That's why it increases the period.

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That's why it takes longer.

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So increasing the period
means it takes longer

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for this thing to go through a cycle,

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and that makes sense in terms of the mass.

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How about this k value?

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That should make sense too.

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If we increase the k value, look it,

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increasing the k would
give us more spring force

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for the same amount of stretch.

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So, if we increase the k value,

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this force from the
spring is gonna be bigger,

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so it can pull harder and
push harder on this mass.

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And so, if you exert a
larger force on a mass,

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you can move it around more quickly,

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and so, larger force means
you can make this mass

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go through a cycle more quickly

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and that's why increasing this
k gives you a smaller period

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because if you can whip this
mass around more quickly,

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it takes less time for
it to go through a cycle

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and the period's gonna be less.

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That confuses people sometimes,

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taking more time means it's
gonna have a larger period.

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Sometimes, people think

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if this mass gets moved around faster,

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you should have a bigger period,

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but that's the opposite.

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If you move this mass around faster,

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it's gonna take less time to move around,

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and the period is gonna decrease

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if you increase that k value.

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So this is what the period of
a mass on a spring depends on.

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Note, it does not depend on amplitude.

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So this is important.

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No amplitude up here.

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Change the amplitude, doesn't matter.

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Those effects offset.

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It only depends on the mass
and the spring constant.

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Again, I didn't derive this.

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If you're curious, watch
those videos that do derive it

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where we use calculus to show this.

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Something else that's important to note,

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this equation works even if
the mass is hanging vertically.

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So, if you have this mass
hanging from the ceiling,

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right, something like this,

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and this mass oscillates
vertically up and down,

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this equation would
still give you the period

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of a mass on a spring.

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You'd plug in the mass that
you had on the spring here.

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You'd plug in the spring
constant of the spring there.

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This would still give you the period

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of the mass on a spring.

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In other words, it does not depend

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on the gravitational constant,

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so little g doesn't show up in here.

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Little g would cause this
thing to hang downward

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at a lower equilibrium point,

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but it does not affect the
period of this mass on a spring,

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which is good news.

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This formula works for horizontal masses,

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works for vertical masses,

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gives you the period in both cases.

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So, recapping, the period
of a mass on a spring

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does not depend on the amplitude.

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You can change the amplitude,

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but it will not affect how
long it takes this mass

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to go through a whole cycle.

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And that's true for
horizontal masses on a spring

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and vertical masses on a spring.

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The period also does not depend on

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the gravitational acceleration,

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so if you took this mass on
a spring to Mars or the moon,

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hung it vertically, let it oscillate,

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if it's the same mass and the same spring,

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it would have the same period.

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It doesn't depend on what the
acceleration due to gravity is

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but the period is affected
by the mass on a spring.

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Bigger mass means you
would get more period

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because there's more inertia,

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and it's also affected
by the spring constant.

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Bigger spring constant
means you'd have less period

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because the force from the
spring would be larger.