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Welcome back.

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So let's do a potential[br]energy problem with

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a compressed spring.

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So let's make this an[br]interesting problem.

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Let's say I have[br]a loop-d-loop.

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A loop-d-loop made out of ice.

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And I made it out of ice so that[br]we don't have friction.

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Let me draw my loop-d-loop.

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There's the loop, there's[br]the d-loop.

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All right.

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And let's say this loop-d-loop[br]has a radius of 1 meter.

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Let's say this is-- this right[br]here-- is 1 meter.

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So of course the loop-d-loop[br]is 2 meters high.

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And let's say I have a[br]spring here-- it's

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a compressed spring.

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Let's say this is the wall.

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This is my spring, it's[br]compressed, so it's

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all tight like that.

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And let's say its spring[br]constant, k, is,

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I don't know, 10.

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Attached to that compressed[br]spring-- so I have a block of

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ice, because I need ice on ice,[br]so I have no friction.

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This is my block of[br]ice, shining.

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And let's say the block of ice[br]is, I don't know, 4 kilograms.

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And we also know that we are[br]on Earth, and that's

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important, because this problem[br]might have been

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different if we were[br]on another planet.

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And my question to you is how[br]much do we have to compress

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the spring-- so, let's say[br]that the spring's natural

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state was here, right, if[br]we didn't push on it.

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And now it's here.

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So what is this distance?

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How much do I have to compress[br]this spring, in order for when

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I let go of the spring, the[br]block goes with enough speed

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and enough energy, that it's[br]able to complete the

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loop-d-loop, and reach safely[br]to the other end?

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So, how do we do this problem?

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Well, in order-- any loop-d-loop[br]problem, the hard

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part is completing the[br]high point of the

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loop-d-loop, right?

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The hard part is making sure[br]you have enough velocity at

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this point, so that you[br]don't fall down.

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Your velocity has to offset the[br]downward acceleraton, in

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which case-- and here, is going[br]to be the centripetal

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acceleration, right?

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So that's one thing[br]to think about.

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And you might say, wow this is[br]complicated, I have a spring

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here, it's going to accelerate[br]the block.

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And then the block's going to[br]get here, and then it's going

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to decelerate, decelerate.

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This is probably where it's[br]going to be at its slowest,

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then it's going to accelerate[br]back here.

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It's a super complicated[br]problem.

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And in physics, whenever you[br]have a super complicated

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problem, it's probably because[br]you are approaching it in a

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super complicated way,[br]but there might be a

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simple way to do it.

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And that's using energy--[br]potential and kinetic energy.

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And what we learned when we[br]learned about potential and

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kinetic energy, is that the[br]total energy in the system

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doesn't change.

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It just gets converted from[br]one form to another.

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So it goes from potential[br]energy to kinetic

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energy, or to heat.

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And we assume that[br]there's no heat,

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because there's no friction.

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So let's do this problem.

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So what we want to know is, how[br]much do I have to compress

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this spring?

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So what I'm essentially saying[br]is, how much potential energy

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do I have to start off with--[br]with this compressed spring--

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in order to make it up here?

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So what's the potential[br]energy?

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Let's say I compress the[br]spring x meters.

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And in the last video, how[br]much potential energy

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would I then have?

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Well, we learned that the[br]potential energy of a

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compressed spring-- and I'll[br]call this the initial

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potential energy-- the initial[br]potential energy, with an i--

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is equal to 1/2 kx squared.

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And we know what k is.

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I told you that the spring[br]constant for the spring is 10.

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So my initial potential energy[br]is going to be 1/2 times 10,

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times x squared.

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So what are all of the energy[br]components here?

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Well, obviously, at this point,[br]the block's going to

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have to be moving, in order[br]to not fall down.

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So it's going to have[br]some velocity, v.

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It's going tangential[br]to the loop-d-loop.

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And it also is going to have[br]some potential energy still.

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And where is that potential[br]energy coming from?

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Well, it's going to come because[br]it's up in the air.

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It's above the surface[br]of the loop-d-loop.

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So it's going to have some[br]gravitational potential

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energy, right?

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So at this point, we're going[br]to have some kinetic energy.

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We'll call that-- well, I'll[br]just call that kinetic energy

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final-- because this is while[br]we care about alpha, maybe

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here it might be the kinetic[br]energy final, but I'll just

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define this as kinetic[br]energy final.

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And then plus the potential[br]energy final.

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And that of course, has to[br]add up to 10x squared.

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And this, of course, now, this[br]was kind of called the spring

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potential energy,[br]and now this is

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gravitational potential energy.

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So what's the energy[br]at this point?

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Well, what's kinetic energy?

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Kinetic energy final is going[br]to have to be 1/2 times the

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mass times the velocity[br]squared, right?

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And then what's the potential[br]energy at this point?

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It's gravitational potential[br]energy, so it's the mass times

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gravity times this height.

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Right?

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So I'll write that here.

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Potential energy final is going[br]to be mass times gravity

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times the height, which also[br]stands for Mass General

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Hospital, anyway.

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You can tell my wife's[br]a doctor, so my

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brain just-- anyway.

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So let's figure out the kinetic[br]energy at this point.

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So what does the velocity[br]have to be?

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Well, we have to figure out[br]what the centripetal

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acceleration is, and then, given[br]that, we can figure out

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the velocity.

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Because we know that the[br]centripetal acceleration-- and

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I'll change colors for[br]variety-- centripetal

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acceleration has to be the[br]velocity squared, over the

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radius, right?

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Or we could say-- and what is[br]the centripetal acceleration

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at this point?

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Well it's just the acceleration[br]of gravity, 9.8

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meters per second squared.

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So 9.8 meters per second[br]squared is equal to v

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squared over r.

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And what's the radius[br]of this loop-d-loop?

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Well it's 1.

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So v squared over r[br]is just going to

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be equal to v squared.

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So v squared equals 9.8-- we[br]could take the square root, or

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we could just substitute the[br]9.8 straight into this

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equation, right?

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So the kinetic energy final is[br]going to be equal to 1/2 times

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the mass times 4 times[br]v squared times 9.8.

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And that equals-- let's just use[br]g for 9.8, because I think

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that might keep it[br]interesting.

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So this is just g, right?

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So it's 2 times g.

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So the kinetic energy final[br]is equal to 2g-- and g is

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normally kilogram meters per[br]second squared, but now it's

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energy, right?

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So it's going to be in joules.

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But it's 2g, right?

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And what is the potential[br]energy at this point?

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Well, it's the mass, which is[br]4, times g times the height,

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which is 2.

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So it's equal to 8g.

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Right.

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So what's the total energy[br]at this point?

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The kinetic energy is 2g, the[br]potential energy is 8g, so the

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total energy at this[br]point is 10g.

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10g total energy.

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So if the total energy at this[br]point is 10g, and we didn't

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lose any energy to friction[br]and heat, and all of that.

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So then the total energy[br]at this point has also

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got to equal 10g.

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And at this point we have no[br]kinetic energy, because this

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block hasn't started[br]moving yet.

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So all the energy is[br]a potential energy.

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So this also has to equal 10g.

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And this g, I keep saying,[br]is just 9.8.

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I just wanted to do that just[br]so you see that it's a

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multiple of 9.8, just for[br]you to think about.

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So what do we have here?

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[? I'll do ?]

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these numbers worked out well.

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So let's divide both[br]sides by 10.

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You get x squared is equal[br]to g, which is 9.8.

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So the x is going to be equal to[br]the square root of g, which

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is going to be equal to what?

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Let's see-- if I take 9.8, take[br]the square root of it,

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it's like 3.13.

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So x is 3.13.

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So we just did a fairly-- what[br]seemed to be a difficult

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problem, but it wasn't so bad.

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We just said that, well the[br]energy in the beginning has to

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be the energy at any point in[br]this, assuming that none of

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the energy is lost to heat.

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And so we just figured out[br]that if we compress this

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spring, with the spring[br]constant of 10.

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If we compress it 3.3 meters--[br]3.13 meters-- we will have

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created enough potential[br]energy-- and in this case, the

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potential energy is 10 times[br]9.8, so roughly 98 joules.

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98 joules of potential energy[br]to carry this object all the

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way with enough velocity at the[br]top of the loop-d-loop to

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complete it, and then come[br]back down safely.

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And so if we wanted to think[br]about it, what's the kinetic

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energy at this point?

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Well we figured out it[br]was 2 times g, so

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it's like 19.6 joules.

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Right.

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And then at this point,[br]it is 98 joules.

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Right?

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Did I do that right?

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Well, anyway I'm running out[br]of time, so I hope I did do

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that last part right.

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But I'll see you in[br]the next video.