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Welcome back.

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So let's do a potential
energy problem with

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a compressed spring.

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So let's make this an
interesting problem.

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Let's say I have
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
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
the d-loop.

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All right.

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

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

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

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And let's say I have a
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
compressed, so it's

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all tight like that.

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And let's say its spring
constant, k, is,

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I don't know, 10.

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

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ice, because I need ice on ice,
so I have no friction.

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This is my block of
ice, shining.

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

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

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important, because this problem
might have been

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different if we were
on another planet.

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

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the spring-- so, let's say
that the spring's natural

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state was here, right, if
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
this spring, in order for when

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

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

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loop-d-loop, and reach safely
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
problem, the hard

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part is completing the
high point of the

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loop-d-loop, right?

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The hard part is making sure
you have enough velocity at

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

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Your velocity has to offset the
downward acceleraton, in

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

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acceleration, right?

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So that's one thing
to think about.

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

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here, it's going to accelerate
the block.

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

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to decelerate, decelerate.

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

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then it's going to accelerate
back here.

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It's a super complicated
problem.

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And in physics, whenever you
have a super complicated

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

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super complicated way,
but there might be a

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simple way to do it.

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And that's using energy--
potential and kinetic energy.

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And what we learned when we
learned about potential and

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kinetic energy, is that the
total energy in the system

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doesn't change.

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It just gets converted from
one form to another.

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So it goes from potential
energy to kinetic

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energy, or to heat.

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And we assume that
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
much do I have to compress

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this spring?

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

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do I have to start off with--
with this compressed spring--

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in order to make it up here?

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So what's the potential
energy?

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Let's say I compress the
spring x meters.

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And in the last video, how
much potential energy

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would I then have?

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Well, we learned that the
potential energy of a

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compressed spring-- and I'll
call this the initial

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potential energy-- the initial
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
constant for the spring is 10.

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So my initial potential energy
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
components here?

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

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have to be moving, in order
to not fall down.

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

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It's going tangential
to the loop-d-loop.

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

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And where is that potential
energy coming from?

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

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It's above the surface
of the loop-d-loop.

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So it's going to have some
gravitational potential

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energy, right?

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

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

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

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

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define this as kinetic
energy final.

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And then plus the potential
energy final.

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

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And this, of course, now, this
was kind of called the spring

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potential energy,
and now this is

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gravitational potential energy.

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So what's the energy
at this point?

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Well, what's kinetic energy?

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

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mass times the velocity
squared, right?

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

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It's gravitational potential
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
to be mass times gravity

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times the height, which also
stands for Mass General

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Hospital, anyway.

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You can tell my wife's
a doctor, so my

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brain just-- anyway.

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

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So what does the velocity
have to be?

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Well, we have to figure out
what the centripetal

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acceleration is, and then, given
that, we can figure out

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the velocity.

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Because we know that the
centripetal acceleration-- and

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I'll change colors for
variety-- centripetal

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acceleration has to be the
velocity squared, over the

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radius, right?

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Or we could say-- and what is
the centripetal acceleration

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at this point?

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Well it's just the acceleration
of gravity, 9.8

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meters per second squared.

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So 9.8 meters per second
squared is equal to v

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squared over r.

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

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Well it's 1.

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So v squared over r
is just going to

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be equal to v squared.

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

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we could just substitute the
9.8 straight into this

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equation, right?

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

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the mass times 4 times
v squared times 9.8.

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

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that might keep it
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
is equal to 2g-- and g is

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normally kilogram meters per
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
energy at this point?

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Well, it's the mass, which is
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
at this point?

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The kinetic energy is 2g, the
potential energy is 8g, so the

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total energy at this
point is 10g.

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10g total energy.

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

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

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So then the total energy
at this point has also

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got to equal 10g.

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And at this point we have no
kinetic energy, because this

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block hasn't started
moving yet.

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So all the energy is
a potential energy.

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So this also has to equal 10g.

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And this g, I keep saying,
is just 9.8.

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

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multiple of 9.8, just for
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
sides by 10.

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

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So the x is going to be equal to
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
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
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
energy in the beginning has to

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be the energy at any point in
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
that if we compress this

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spring, with the spring
constant of 10.

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If we compress it 3.3 meters--
3.13 meters-- we will have

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created enough potential
energy-- and in this case, the

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

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98 joules of potential energy
to carry this object all the

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

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complete it, and then come
back down safely.

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

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energy at this point?

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Well we figured out it
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,
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
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
the next video.