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chapter 13 is about simple harmonic
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motion In Waves we'll talk about simple
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harmonic motion first simple harmonic
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motion is a subtype of periodic motion
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which means an object repeating the same
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path over and over taking the same
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amount of time which is called the
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period capital T each cycle
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we already saw one example of periodic
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motion which is uniform circular motion
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like the ball on an end of a string
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whirling around in a circle with
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constant speed or planets orbiting
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around their star that's one example of
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periodic motion repeating the same orbit
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over and over
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other examples of periodic motion
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include Springs and pendulums and those
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are a special subtype of periodic motion
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that are called simple harmonic motion
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simple harmonic motion is just periodic
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motion that means for two further
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criteria Criterion one is that the force
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needs to be proportional to the
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displacement Criterion two is that the
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force is always directed back toward the
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equilibrium position meaning that it is
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a restoring Force
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Springs are only simple harmonic
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oscillators meaning things doing simple
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harmonic motion if there's no friction
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there
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in the real world where there's friction
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if I were to pull the this Mass downward
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and then let it go it would boing back
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and forth back and forth but the
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amplitude of its motion would decrease
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over time until it eventually came back
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to rest that would be the influence of
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friction and we call that a damped
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harmonic oscillator a simple harmonic
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oscillator in a system that has no
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friction the motion would not damp the
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amplitude would stay the same forever
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and the object would repeat the exact
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same path over and over and over and
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over again with the same amount of time
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per cycle for each one
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similarly for a pendulum if we assume
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that there is no friction up at the
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pivot and no air resistance if we set
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our pendulum at some initial
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displacement called the amplitude and
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then let it go it's going to go tick
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tock tick tock back and forth with the
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same amplitude of motion over and over
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and over again taking the same amount of
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time for each cycle
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Springs and pendulums are not the only
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examples of simple harmonic oscillators
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simple harmonic motion occurs across
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many different types of physics problems
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you'll see simple harmonic oscillators
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come back in quantum mechanics and
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condensed matter physics and
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astrophysics and all kinds of sort of
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more complex physical situations
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and so the beauty of simple harmonic
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motion is that the equations that we're
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going to derive in this chapter for a
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relatively simple systems like a spring
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and a pendulum would be ones that you
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can still use as like a physics grad
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student studying a much more complex
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problem
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let's talk about Springs first this is a
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slide from chapter five where we first
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learned spring Force equation which is
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called hooke's law and it says that the
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spring Force has a vector is equal to
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negative K times the displacement X as a
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vector so the negative here is telling
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us that the spring force and
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displacement are in opposite directions
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from each other so if we have an Vector
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X the displacement that is pointing to
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the right because someone pulled the
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spring to the right the spring force is
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to the left notice that's pointing back
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towards equilibrium so that's meeting
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the second Criterion about simple
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harmonic motion we've got a restoring
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Force right here
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the other Criterion the first Criterion
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for simple harmonic motion was that the
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force was proportional to the
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displacement and we can in fact here see
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that our spring force is proportional to
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X and so Springs therefore are going to
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be examples of simple harmonic
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oscillators as long as the spring force
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is the only force that is acting
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which means we have to assume no
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friction
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the other thing that we might need to
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ignore is gravity so let's think about
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the example of a vertical spring
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if we have a vertical spring hanging
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here with no Mass on the end it's just
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going to be hanging there with some
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length to it the end of the spring we
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would say is the old equilibrium
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position
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now if we attach a mass to the end of
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the spring the spring is going to
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stretch a bit and now the end of the
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spring we would say is that the new
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equilibrium position
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the distance that the spring stretched
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let's call X naught
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if the mass is in equilibrium here on
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the end of the spring that means our
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Mass is at rest staying at rest
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um for example then that means that the
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net force is going to be equal to zero
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our spring Force pulling up is equal to
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K times x naught our Gravity Force
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pulling down is mg so k x naught minus
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mg equals zero and we can solve for x
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naught
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this tells us how much the spring would
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stretch when we put some Mass M on it
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if now we Define our new equilibrium
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position to be the new x equals zero
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spot
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when we pull the mass downward and Let
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It Go the oscillation of the spring is
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going to be centered around that new
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equilibrium position the new x equals to
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zero and the gravity is no longer
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important so gravity just told us how
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much to sort of redefine our coordinate
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system here but now we don't need to
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consider gravity as a force acting on
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the mass anymore we've already taken
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that into account
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and that means we can still treat
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vertical Springs as simple harmonic
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oscillators too as long as there's no
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friction
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if the spring force is the only force in
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the problem because we have already
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accounted for the gravity if it's a
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vertical Spring by redefining the
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equilibrium position and we're saying
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there's no friction then our spring
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force and net force are the same
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spring force is equal to negative KX net
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force according to Newton's second law
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of motion is equal to mass times
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acceleration setting those equal to each
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other lets us solve for the acceleration
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of a mass spring system
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and you can see here that the
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acceleration depends on displacement
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so if we have a spring that is boinging
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back and forth with X changing as a
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function of time that means a is also
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going to end up changing as a function
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of time that the acceleration is not a
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constant and that means we can't use
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kinematic equations here which were only
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valid for constant acceleration and
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we're going to have to come up with a
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set of new equations to use in place of
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kinematics equations
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satisfying the same sort of need of
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being able to relate position velocity
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acceleration and time to each other
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we've already got a start on that this
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equation relates our acceleration and
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our displacement which would be the same
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thing as the position of the end of the
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spring to each other but now we also
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want something that's going to relate
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um our velocity to those things and then
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also to time and so we'll need to sort
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of build up those equations
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here's an illustration showing how
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acceleration and speed are depending on
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displacement and then since displacement
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changes with time that means they also
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are going to end up depending on time
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imagine that we had it block on the end
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of a spring that was displaced to the
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right initially and then let go
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if the block was held at rest up until
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it was let go our initial speed is going
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to be zero
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and there's an acceleration that is to
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the left in the opposite direction as
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our displacement Vector which is
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pointing to the right since the block
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got pulled to the right before being let
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go
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now as our acceleration is to the left
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that's going to cause the block to start
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moving to the left
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when the block reaches equilibrium
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there's no more acceleration at that
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point but there is already a speed to
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the left so the block is going to
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continue moving through equilibrium past
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equilibrium it keeps moving left but now
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our acceleration is to the right
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pointing back towards equilibrium
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because we have a restoring force uh and
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so our block is moving left while
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accelerations to the right we get a
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slowing down that happens until
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eventually the block is going to come to
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rest momentarily so V will be zero just
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for a moment but at that point the
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acceleration is in zero our acceleration
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is to the right so the block doesn't
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stay at rest but instead starts moving
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to the right
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it passes through equilibrium again so
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we've got another moment where a is zero
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but the speed is maximum and then it
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keeps going until it stops again
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momentarily at the displacement over
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here on the other side and when we reach
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a maximum displacement that is equal to
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the amplitude
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since we're saying no friction the spot
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it gets to here has the same
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displacement a as it did at the very
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beginning
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so back and forth back and forth you can
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see that the acceleration and the speed
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are both continuously changing during
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this motion we're going to need to get
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an equation that will give us V as a
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function of X just like this one is a as
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a function of x
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and then we're also going to want
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equations that give us V and a and X as
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functions of time and so that will be in
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the next videos deriving those equations
