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PHYSICS 4A EQUATION SHEET Form: Customize, Finish and save

now what we are going to do is very.actually going to do the hard work and.do the math so this is if you wanna.follow along in the worksheet this is in.sort of the next section starting on.page three but let me give you this.description so what we are trying to do.is we are trying to analyze this setup.so completely that we can describe the.position of the mass as a function of.time this is once again.at a level of detail that we haven't.done before before we did as far as you.know figuring out what is the amount of.force when I release it here so we've.done that we have looked at snapshots.you know what is the position and.velocity here what is the position and.velocity here and we kind of didn't.really do it but you could have figured.that position and velocity at any of.these points like if it's at this point.then you to figure out what is the.potential energy and you do use that to.figure out kinetic energy and find the.velocity but one thing that we have.never done is we have never described it.as a function of time yeah so let's just.see how we can do it so let me title.this analyzing and I'm just going to.give this a name this particular type of.motion that you see it has a name it's a.very specific type of thing that you.will see a lot over the course of your.science and engineering courses we call.it simple harmonic oscillator motion so.analyzing a simple harmonic oscillator.yes so let's just start off with this.picture let's say I have some mass and.that's hanging from a spring I have to.start specifying some of the parameters.here let's say that the spring constant.is K and I guess I can say.that this is my equilibrium position so.this is x equals zero all right and this.is a question that I can pose an answer.let's say I pull this down a little bit.so I pull this mess down to down over.here and so you know this spring sort of.comes and message down here I pulled it.down to some initial position X is equal.to X naught and then I let go and the.question that we are now going to try to.answer is what is the position as a.function of time.any ideas how you do begin to approach.this kind of question doesn't quite fit.into any of the boxes that we have.talked about it's because you know if.you conserve a sheen of energy they do.not give anything as a function of time.right but you know we have to start.somewhere this is a type of problem you.haven't seen before so you know more.detailed and complicated in anything you.have seen before we are saying we have.to start up from somewhere to try to.figure this out where would you start.the starting point is actually not old.and you would it surprise you if I said.that in trying to answer this question.we actually start from standard strategy.like does that sound like a method that.would work maybe it's not completely.irrelevant right because one of the.results that you know you can get from.standard strategy is you can get.acceleration right and you know that.acceleration is related to position you.know acceleration is rate of change of.velocity which is itself rate of change.of position so acceleration is the.double time derivative position like.it's not as though it's an unrelated so.far we have tried to avoid doing this.because then you know it gets.complicated.but now this is a new situation there's.no other way to really handle it so we.are going to start off with the standard.strategy and what I want to really.impress you on is any kind of problem.you see in mechanic's the analysis of.force it's something that actually it.always works it's just that sometimes.you have to use some brute force because.the expression you get is not simple you.might have to do numerical calculation.with a computer this happens to be a.situation where we can actually solve it.exactly but so you know if you'll see a.new problem and you don't know how to.approach it you can always start by.analyzing the forces that's why we start.this class that's why we start your.sequence of physics classes with the.forces because that's your fallback.method for everything so let's just.start from this so if you're doing.standard strategy then we have to start.off with a free body diagram so we have.a free body diagram.let me cheat a little bit here I'm going.to switch back to describing the forces.on this mass as this quote-unquote.spring force as in I am going to.describe one force that is a combination.of gravity and spring force and I'm.going to call that my spring force there.used to be doing this right yes so right.now at this position what would you say.is the spring force that's the pure.spring force not the one with the.quotation mark around it what would you.say the spring force here is so mg plus.KX with the directions in mind what.should that be I reckon I thought you.said an answer yeah zero it's at.equilibrium so this is what we are.treating as the equilibrium position of.the codon foot spring at equilibrium.portion of the spring the spring force.is zero so yeah spring force is zero.here so imagine this situation where we.pulled it down what's the direction of.spring force upward and so let me draw.that so here's the mass so I have a.spring force upward and what would you.say is the magnitude of the spring force.in terms of in terms of all the.parameters that's written on the board.what is the magnitude of this well I.have to blackness what is the magnitude.of this hold on called spring force.magnitude is K it is K times X not this.position displacement from equilibrium.so the matter that you're going to be K.times X not all right let me introduce.one more so those notation.you know convention thing and we will.finish this portion and I guess we have.to take break and then we'll finish it.up this description in a bit from the.way you've submitted things so far which.it direction would you say I'm which.direction is positive am I so you know.if you've seen this written down so far.and I asked you what is the positive.direction would you say that upward is.positive or downward is positive so a.lot of people do intuitively choose.upward is positive since I don't want to.go against your intuition so let me.correct a couple of things X naught it's.kind of natural to pick it as a positive.number that means for my X I really.should have called it minus X naught.because I pulled it downward right so.okay so my not fine there and for this.force so right now when the mass is down.here the force is pointing upward now.when I let go I want you to imagine just.any general position in this.neighborhood is the force always going.to be pointing upward so now right.sometimes it's going to point downward.so I want to write down this expression.so that I'm not talking about just the.magnitude I'm actually talking about the.force as a vector quantity with the.direction in mind so I want to have a.description for the force with where the.sign plus or minus signs are important.as a general function of X is this okay.or do I need to change something as in.for this position if I say the force as.a vector is K times X it's okay.presumably a positive quantity times X.is K times X right now describing an.upward force or downward force.describing downward force right because.X is a negative number I don't want that.for this position I want my force to be.positive upward so what can I fix to.make my force upward right heard someone.mumble something there and here but I.didn't hear any of the Argent were you.saying something.make the X negative okay I can do that.so you mean something like a x minus X.now actually that's all I think that's.actually conceptually cleaner so do you.have actually seen it in this format you.have seen this as minus KX and I don't.know if I I don't think I mentioned this.name before this particular form of.force.it's called restoring force if you are.reading a textbook you should have seen.this phrase at some point and the reason.that this is called a restoring force is.for the reason that Arjun was describing.the direction of force is negative the.opposite to the displacement so so I.could just you know force is equal to.minus K times X as a vector where I care.about the direction and let's let's make.sure that that makes a sense when it's.on the other side of the equilibrium so.when it's above the equilibrium is that.my X positive or negative positive is my.force negative then ya downward right.all right so all of this sounds fine so.all right so any other forces on this.mess just this quote-unquote spring.force that's actually a combination of.spring plus gravity right that's the.only force on this mess okay then I.think I'm done with all the standard.straightish steps and I'm ready to write.my Newton's second law equation so.Newton's second law says that net force.in this case it's just the spring force.once again cause I'm called spring force.so my spring force is equal to mass.times acceleration now so from this I.can write this down that my spring force.or minus KX is equal to mass times.acceleration and you know this is.nothing that you haven't seen before and.what I am now going to do is something.that you have not seen before I'm not.going to stop here just satisfied with.no video knowing acceleration because.that's not my goal this time my goal.this time is to get the position as a.function of time so I can't stop here I.have to try to somehow Express this to.put equation in terms of nothing but X.so you know your theory for a while and.you realize well there's no algebraic.way to do it so the only way you can do.it is through calculus you write down.what acceleration is in terms of.position so you do that one more step.and say it's mass times double time.derivative of position so I don't.sometimes I'm a little bit sloppy about.when I indicate this or not so once I.start doing this what you have to.implicitly remind yourself every single.time is that position is a function of.time that's the only way this derivative.makes sense that what that means is that.this position that I have written here.it's a function of time it's not a.simple number it's an entire function.okay let me clean this up a little bit.let me solve this is the most.complicated looking thing in the entire.equation so let me solve for this so.solving for that I get this double time.derivative of position is equal to let's.move the mess over so it's minus K over.m times X and once again it's one last.to remind.at some point I'm going to get lazy and.not remind to you at some point X is a.function of time both the here and here.so this is a this is what we call.differential equation how many here have.seen a differential equation in your.math class you should have seen it if.you haven't solved one so I'm we call it.differential equation because it's a.equation that involves derivatives.here's here's my oops forgot so here.here's my derivative it's not any.derivative it's a second order.derivative so it's a differential.equation involving second order.derivative time and in the context of.physics this equation actually has a.name we call this equation of motion the.idea here is that this equation once.written down completely determines how.something moves so this is the equation.of motion.once you have equation of motion for.something then it's you can determine.what it's going to do for so if you know.the equation of motion and if you know.where it was at a particular time then.for all future times you can determine.what it's going to do I mean if you're.into philosophy there's some philosophy.called implication to that but I won't.get into that this is lower division.physics so this is the equation of.motion and the reason we are introducing.this only now is well this is kind of.difficult to solve for it's not.something you can solve for.algebraically but we'll try to solve for.it when we come back from break let's.take the break now.

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