Harmonic Motion Part 3 (no calculus) Figuring out the period, frequency, and amplitude of the harmonic motion of a mass attached to a spring.
Harmonic Motion Part 3 (no calculus)
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- Welcome back.
- And if you were covering your eyes because you didn't want
- to see calculus, I think you can open your eyes again.
- There shouldn't be any significant displays of
- calculus in this video.
- But just to review what we went over, we just said, OK if
- we have a spring-- and I drew it vertically this time-- but
- pretend like there's no gravity, or maybe pretend like
- we're viewing-- we're looking at the top of a table, because
- we don't want to look at the effect of
- a spring and gravity.
- We just want to look at a spring by itself.
- So this could be in deep space, or something else.
- But we're not thinking about gravity.
- But I drew it vertically just so that we can get more
- intuition for this curve.
- Well, we started off saying is if I have a spring and 0-- x
- equals 0 is kind of the natural resting point of the
- spring, if I just let this mass-- if I didn't pull on the
- spring at all.
- But I have a mass attached to the spring, and if I were to
- stretch the spring to point A, we said, well what happens?
- Well, it starts with very little velocity, but there's a
- restorative force, that's going to be pulling it back
- towards this position.
- So that force will accelerate the mass, accelerate the mass,
- accelerate the mass, until it gets right here.
- And then it'll have a lot of velocity here, but then it'll
- start decelerating.
- And then it'll decelerate, decelerate, decelerate.
- Its velocity will stop, and it'll come back up.
- And if we drew this as a function of time,
- this is what happens.
- It starts moving very slowly, accelerates.
- At this point, at x equals 0, it has its maximum speed.
- So the rate of change of velocity-- or the rate of
- change of position is fastest. And we can see the slope is
- very fast right here.
- And then, we start slowing down again, slowing down,
- until we get back to the spot of A.
- And then we keep going up and down, up and down, like that.
- And we showed that actually, the equation for the mass's
- position as a function of time is x of t-- and we used a
- little bit of differential equations to prove it.
- But this equation-- not that I recommend that you memorize
- anything-- but this is a pretty
- useful equation to memorize.
- Because you can use it to pretty much figure out
- anything-- about the position, or of the mass at any given
- time, or the frequency of this oscillatory motion, or
- anything else.
- Even the velocity, if you know a little bit of calculus, you
- can figure out the velocity at anytime, of the object.
- And that's pretty neat.
- So what can we do now?
- Well, let's try to figure out the period of
- this oscillating system.
- And just so you know-- I know I put the label harmonic
- motion on all of these-- this is simple harmonic motion.
- Simple harmonic motion is something that can be
- described by a trigonometric function like this.
- And it just oscillates back and forth, back and forth.
- And so, what we're doing is harmonic motion.
- And now, let's figure out what this period is.
- Remember we said that after T seconds, it gets back to its
- original position, and then after another T seconds, it
- gets back to its original position.
- Let's figure out with this T is.
- And that's essentially its period, right?
- What's the period of a function?
- It's how long it takes to get back to your starting point.
- Or how long it takes for the whole cycle to happen once.
- So what is this T?
- So let me ask you a question.
- What are all the points-- that if this is a
- cosine function, right?
- What are all of the points at which cosine is equal to 1?
- Or this function would be equal to A, right?
- Because whenever cosine is equal to 1, this whole
- function is equal to A.
- And it's these points.
- Well cosine is equal to 1 when-- so, theta-- let's say,
- when is cosine of theta equal to 1?
- So, at what angles is this true?
- Well it's true at theta is equal to 0, right?
- Cosine of 0 is 1.
- Cosine of 2 pi is also 1, right?
- We could just keep going around that unit circle.
- You should watch the unit circle video if this makes no
- sense to you.
- Or the graphing trig functions.
- It's also true at 4 pi.
- Really, any multiple of 2 pi, this is true.
- Cosine of that angle is equal to 1.
- So the same thing is true.
- This function, x of t, is equal to A at what points?
- x of t is equal to A whenever this expression-- within the
- cosines-- whenever this expression is equal to 0, 2
- pi, 4 pi, et cetera.
- And this first time that it cycles, right, from 0 to 2
- pi-- from 0 to T, that'll be at 2 pi, right?
- So this whole expression will equal A, when k-- and that's
- these points, right?
- That's when this function is equal to A.
- It'll happen again over here someplace.
- When this little internal expression is equal to 2 pi,
- or really any multiple of 2 pi.
- So we could say, so x of t is equal to A when the square
- root of k over m times t, is equal to 2 pi.
- Or another way of thinking about it, is let's multiply
- both sides of this equation times the inverse of the
- square root of k over m.
- And you get, t is equal to 2 pi times the square root-- and
- it's going to be the inverse of this, right?
- Of m over k.
- And there we have the period of this function.
- This is going to be equal to 2 pi times the square
- root of m over k.
- So if someone tells you, well I have a spring that I'm going
- to pull from some-- I'm going to stretch it, or compress it
- a little bit, then I let go-- what is the period?
- How long does it take for the spring to go back to its
- original position?
- It'll keep doing that, as we have no friction, or no
- gravity, or any air resistance, or
- anything like that.
- Air resistance really is just a form of friction.
- You could immediately-- if you memorize this formula,
- although you should know where it comes from-- you could
- immediately say, well I know how long the period is.
- It's 2 pi times m over k.
- That's how long it's going to take the spring to get back--
- to complete one cycle.
- And then what about the frequency?
- If you wanted to know cycles per second, well that's just
- the inverse of the period, right?
- So if I wanted to know the frequency, that equals 1 over
- the period, right?
- Period is given in seconds per cycle.
- So frequency is cycles per second, and this
- is seconds per cycle.
- So frequency is just going to be 1 over this.
- Which is 1 over 2 pi times the square root of k over m.
- That's the frequency.
- But I have always had trouble memorizing this, and this.
- You always [UNINTELLIGIBLE]
- k over m, and m over k, and all of that.
- All you have to really memorize is this.
- And even that, you might even have an intuition
- as to why it's true.
- You can even go to the differential equations if you
- want to reprove it to yourself.
- Because if you have this, you really can answer any question
- about the position of the mass, at any time.
- The velocity of the mass, at any time, just by taking the
- Or the period, or the frequency of the function.
- As long as you know how to take the period and frequency
- of trig functions.
- You can watch my videos, and watch my trig videos, to get a
- refresher on that.
- One thing that's pretty interesting about this, is
- notice that the frequency and the period, right?
- This is the period of the function, that's how long it
- takes do one cycle.
- This is how many cycles it does in one second-- both of
- them are independent of A.
- So it doesn't matter, I could stretch it only a little bit,
- like there, and it'll take the same amount of time to go
- back, and come back like that, as it would if I
- stretch it a lot.
- It would just do that.
- If I stretched it just a little bit, the function would
- look like this.
- Make sure I do this right.
- I'm not doing that right.
- Edit, undo.
- If I just do it a little bit, the amplitude is going to be
- less, but the function is going to essentially do the
- same thing.
- It's just going to do that.
- So it's going to take the same amount of time to complete the
- cycle, it'll just have a lower amplitude.
- So that's interesting to me, that how much I stretch it,
- it's not going to make it take longer or less time to
- complete one cycle.
- That's interesting.
- And so if I just told you, that I actually start having
- objects compressed, right?
- So in that case, let's say my A is minus 3.
- I have a spring constant of-- let's say k is,
- I don't know, 10.
- And I have a mass of 2 kilograms. Then I could
- immediately tell you what the equation of the position as a
- function of time at any point is.
- It's going to be x of t will equal-- I'm running out of
- space-- so x of t would equal-- this is just basic
- subsitution-- minus 3 cosine of 10 divided by 2, right? k
- over m, is 5.
- So square root of 5t.
- I know that's hard to read, but you get the point.
- I just substituted that.
- But the important thing to know is this-- this is, I
- think, the most important thing-- and then if given a
- trig function, you have trouble remembering how to
- figure out the period or frequency-- although I always
- just think about, when does this expression equal 1?
- And then you can figure out-- when does it equal 1, or when
- does it equal 0-- and you can figure out its period.
- If you don't have it,
- you can memorize this formula for period, and this formula
- for frequency, but I think that might be a waste of your
- brain space.
- Anyway, I'll see you in the next video.
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