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Now we come to the second law.
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Newton's second law.
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I have a spring –
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forget gravity for now,
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you can do this somewhere in outer space –
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this is the relaxed length of the spring,
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and I extend the spring …
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I extend it over a certain amount –
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certain distance – in unimportant how much.
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And I know, that when I do that
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that there will be a pull.
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Non-negotiable.
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I put a mass m₁ here
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and I measure the acceleration
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that this pull causes on this mass
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immediately after I release –
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I can measure that.
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So I measure an acceleration a₁.
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Now I replace this object
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by mass m₂,
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but the extension is the same.
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So the pull must be the same,
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the spring doesn't know
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what the mass is at either end, right?
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So the pull is the same,
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I put m₂ there – different mass –
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and I measure
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new acceleration a₂.
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It is now an experimental fact, that
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m₁ a₁ = m₂ a₂.
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And this product,
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m a, we call the force.
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That is our definition of force.
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So the same pull
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on a ten times larger mass
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would give a
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ten times lower acceleration.
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The second law, I will read to you:
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“A force action on a body
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gives it an acceleration
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which is in the direction of the force […]” –
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that's also important,
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acceleration is in the direction of the force –
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“[…] and has a magnitude given by m a.”
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m a is the magnitude.
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And the direction is
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the direction of the force.
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So now we will write this,
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in all glorious,
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detailed is this the second law
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by Newton;
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Perhaps the most important law
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in <i>all</i> of physics,
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and certainly in all of 8.01.
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F ⃗ = m a ⃗.
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The units of this force
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are kilograms times meters per second squared –
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in honor of the great man
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we call that one Newton.
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Like the first law the second law
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<i>only</i> holds in inertial reference frames.
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Can the first law, the second law be proven?
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No.
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Do we believe in it?
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Yes.
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Why do we believe in it?
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Because <i>all</i> experiments
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and <i>all</i> measurements
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within the uncertainty of the measurements
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are in agreement with the second law.
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Now you may object.
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And you may say:
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“This is strange what you've been doing.”
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“How can you ever determine
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a mass
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if there is no force somewhere?”
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Because if you wanna determine the mass
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maybe you put it on a scale
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and when you put it on the scale
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to determine the mass
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you make use of the gravitational force;
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So, isn't that some kind of a
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circular argument that you're using?
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And your answer is: No.
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I can be somewhere in outer space
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where there's no gravity.
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I have two pieces of cheese
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that are <i>identical</i> in size –
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this is cheese without holes, by the way.
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They are identical in size.
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The sum of the two
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has double the mass of one.
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Mass is determined by
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how many molecules
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how many atoms that I have.
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I don't need gravity
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to have a relative scale of masses
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so I can determine
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the relative scale of these masses
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without ever using a force.
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So, this is a very legitimate way of …
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… checking up on the second law.
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[video glitch]
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Since all objects
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in this lecture hall
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and the earth
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fall is the
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constant acceleration, which is g,
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we can write down
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that
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the gravitational force
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would be
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m times this acceleration g ⃗.
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Normally I write an a ⃗ for it,
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but I make an exception now,
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because gravity I call it
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gravitational force.
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And so you see
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that the gravitational force
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due to the earth
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on a particular mass
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is [unintelligible] proportional
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with the mass
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if the mass becomes ten times larger
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then the force
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due to gravity
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goes up by a factor of ten.
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Suppose I have here
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this softball in my hand.
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In the reference frame
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26.100, [unintelligible] we will accept
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to be an inertial reference frame.
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Is not being accelerated
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in our reference frame.
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That means,
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the force on it must be zero.
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So here
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is that ball.
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And we know, if it has mass m,
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which is in this case is about half a kilogram,
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that there must be a force here,
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m g,
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which is about
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five Newton,
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or half a kilogram.
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But the net force
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is zero.
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Therefore,
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it is very clear
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that I, Walter Lewin,
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must push up
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with a force
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from my hand onto the ball,
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which is about
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or as the same, which is exactly the same
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five Newtons.
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Only now is there
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no acceleration.
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So I can write down, that
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force of Walter Lewin
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plus the force of gravity
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equal zero,
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because it's a one-dimensional problem
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you could say, that
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the force of Walter Lewin
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equals minus m g.
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F ⃗ = m a ⃗.
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Notice,
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that there's no statement made
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on velocity nor speed.
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As long as you know F ⃗,
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and as long as you know m,
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a ⃗ is uniquely specified.
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<i>No</i> information is needed on the speed.
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But that would mean,
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if we take gravity
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and an object was falling down
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with five meters per second,
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that the law would hold.
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If it would fall down
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with 5,000 meters per second,
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it would <i>also</i> hold.
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Will it <i>always</i> hold?
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No.
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Once
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your speed
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approaches the speed of light,
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then it would only mechanics no longer works,
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then you have to use
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Einstein's theory of special relativity.
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So this is only valid
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as long as we have
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speeds that are
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substantially smaller,
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say, than the speed of light.
