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LEWIN: You did very well on your first exam.

I was hoping for an average of about 75; the class average was 89.

So that leaves us with two possibilities: either you are very smart, this is an exceptional class, or the exam was too easy.

Now, this exam was taken by three instructors way before you took it.

None of them thought it was too easy, so I'd like to think that you are really an exceptional class and I'd like to congratulate you that you did so well.

Here is a histogram of the scores.

If we had to decide on this test alone —

forgetting your quizzes, forgetting your homework, on this test alone —

the dividing line between pass and fail would be 65.

That means that five percent of the class would fail, which is unusually low.

Normally that is around 15%.

But time will tell whether you are indeed exceptionally smart or whether the exam was too easy.

The good news also is —

two pieces of good news —

that we promise that the books will arrive at the Coop today.

Today we're going to talk about springs, about pendulums and about simple harmonic oscillators —

one of the key topics in 801.

If I have a spring...

and this is the relaxed length of the string... spring, I call that x equals zero.

And I extend the string...

the spring, with a "p," then there is a force that wants to drive this spring back to equilibrium.

And it is an experimental fact that many springs —

we call them ideal springs —

for many springs, this force is proportional to the displacement, x.

So if this is x, if you make x three times larger, that restoring force is three times larger.

This is a one-dimensional problem, so to avoid the vector notation, we can simply say that the force, therefore, is minus a certain constant, which we call the spring constant —

this is called the spring constant —

and the spring constant has units newtons per meter.

So the minus sign takes care of the direction.

When x is positive, then the force is in the negative direction; when F is negative, the force is in the positive direction.

It is a restoring force.

Whenever this linear relation between F and x holds, that is referred to as Hooke's Law.

How can we measure the spring constant? That's actually not too difficult.

I can use gravity.

Here is the spring in its relaxed situation.

I hang on the spring a mass, m, and I make use of the fact that gravity now exerts a force on the spring, and when you find your new equilibrium —

this is the new equilibrium position —

then the spring force, of course, must be exactly the same as mg.

No acceleration when the thing is at rest.

And so I could now make a plot whereby I could have here x and I could have here this force F, which I know because I know the masses.

I can change the masses.

I can go through a whole lot of them.

And you will see data points which scatter around a straight line.

And the spring constant follows, then, if you take...

if you call this delta F and you call this delta x, then the spring constant, k, is delta F divided by delta x.

So you can even measure it.

You don't have to start necessarily at this point where the spring is relaxed.

You could already start when the spring is already under tension.

That is not a problem.

You'll be surprised how many springs really behave very nicely according to Hooke's Law.

Uh, I have one here.

It's not a very expensive spring.

You see it here.

I was hoping for an average of about 75; the class average was 89.

So that leaves us with two possibilities: either you are very smart, this is an exceptional class, or the exam was too easy.

Now, this exam was taken by three instructors way before you took it.

None of them thought it was too easy, so I'd like to think that you are really an exceptional class and I'd like to congratulate you that you did so well.

Here is a histogram of the scores.

If we had to decide on this test alone —

forgetting your quizzes, forgetting your homework, on this test alone —

the dividing line between pass and fail would be 65.

That means that five percent of the class would fail, which is unusually low.

Normally that is around 15%.

But time will tell whether you are indeed exceptionally smart or whether the exam was too easy.

The good news also is —

two pieces of good news —

that we promise that the books will arrive at the Coop today.

Today we're going to talk about springs, about pendulums and about simple harmonic oscillators —

one of the key topics in 801.

If I have a spring...

and this is the relaxed length of the string... spring, I call that x equals zero.

And I extend the string...

the spring, with a "p," then there is a force that wants to drive this spring back to equilibrium.

And it is an experimental fact that many springs —

we call them ideal springs —

for many springs, this force is proportional to the displacement, x.

So if this is x, if you make x three times larger, that restoring force is three times larger.

This is a one-dimensional problem, so to avoid the vector notation, we can simply say that the force, therefore, is minus a certain constant, which we call the spring constant —

this is called the spring constant —

and the spring constant has units newtons per meter.

So the minus sign takes care of the direction.

When x is positive, then the force is in the negative direction; when F is negative, the force is in the positive direction.

It is a restoring force.

Whenever this linear relation between F and x holds, that is referred to as Hooke's Law.

How can we measure the spring constant? That's actually not too difficult.

I can use gravity.

Here is the spring in its relaxed situation.

I hang on the spring a mass, m, and I make use of the fact that gravity now exerts a force on the spring, and when you find your new equilibrium —

this is the new equilibrium position —

then the spring force, of course, must be exactly the same as mg.

No acceleration when the thing is at rest.

And so I could now make a plot whereby I could have here x and I could have here this force F, which I know because I know the masses.

I can change the masses.

I can go through a whole lot of them.

And you will see data points which scatter around a straight line.

And the spring constant follows, then, if you take...

if you call this delta F and you call this delta x, then the spring constant, k, is delta F divided by delta x.

So you can even measure it.

You don't have to start necessarily at this point where the spring is relaxed.

You could already start when the spring is already under tension.

That is not a problem.

You'll be surprised how many springs really behave very nicely according to Hooke's Law.

Uh, I have one here.

It's not a very expensive spring.

You see it here.

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