WEBVTT
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Welcome back to recitation.
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In this video I'd
like us to work
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on the following application
for spherical coordinates.
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So what I'd like
us to do is find
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the gravitational attraction
of an upper solid half
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sphere of radius a
and center O, that's
00:00:22.700 --> 00:00:26.670
the gravitational attraction
on a mass m naught at O.
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So the m naught is sitting at
O, we have a solid half sphere
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of radius a and
center O, and we want
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to find the gravitational
attraction of the half sphere
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on the mass.
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And we're going to
assume that the density
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delta is equal to
the square root of x
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squared plus y squared.
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So why don't you pause the
video, take a little while
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to work on it, and when you
feel good about your solution,
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bring the video back up,
I'll show you what I did.
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OK.
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Welcome back.
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Again what we're going
to do is we're going
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to do an application problem.
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And what we have is we
have the solid half sphere
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of radius a and
center O, and we want
00:01:04.250 --> 00:01:06.660
to know the gravitational
attraction of it
00:01:06.660 --> 00:01:08.394
on a mass m naught.
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So the first thing
I'm going to do
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is draw a picture, just so
I can get myself oriented.
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The easiest thing
to do-- obviously,
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when I'm going to try
and figure this out,
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I'm going to use
spherical coordinates.
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I have an upper
solid half sphere,
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so that should tell me spherical
coordinates will be good.
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And the easiest way to deal
with spherical coordinates
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is obviously to choose
O to be the origin.
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So I'm going to have m naught
sitting right at the origin.
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And then I'm going
to have-- oops,
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I should write that
somewhere else.
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Maybe, well, I'll
just leave it there--
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and then I'm going to
have the solid half
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sphere, which I'm going to
just draw sort of like this.
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Maybe that doesn't look
like a sphere to you.
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But we have the
solid half sphere
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and it's exerting a
force on m naught.
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And if I want to
know the force-- how
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we denote the force in
lecture was the components
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were the x-component
of the force,
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the y-component of the
force, and the z-component
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of the force.
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And so I want to point out these
are not partial derivatives.
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This is the standard notation
you use for force in this case,
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and so these are not
the partial derivatives.
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They're just the
x-component, the y-component,
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and the z-component.
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And if you notice,
this upper solid half
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sphere-- because of
where m naught is,
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the x-component and the
y-component of the force
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are 0 based on symmetry.
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So all I really need to worry
about is the z-component.
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I'm only really interested
in one component of the force
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and it's the z-component.
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Because the other two
are going to be 0, just
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by the symmetry of
this solid half sphere.
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So I really only
need the z-component.
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So what I have to do, in
order to calculate the force,
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is I have to figure out the
magnitude and the direction.
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So let's write down
the two components.
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These were given
to you in class.
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So the magnitude is going
to be equal to G m naught
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dm divided by rho squared.
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So this was given to
you in class already.
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So G is some constant; m
naught is the mass you have;
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dm, we have to determine
a little bit about it;
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and then we're dividing
by rho squared.
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And the direction--
because I'm only
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interested in the z-component--
the direction is just z,
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but then I want to make sure
that the direction component is
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z divided by rho.
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The full direction is
[x, y, z] divided by rho,
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but I'm only interested
in that last component.
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So the z-component
is just z over rho.
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If that confused
you, I just want
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to remind you that
you had [x, y, z]
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over rho is the full
direction of the force,
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we're just pulling
off the z-component.
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So what we're going to
be trying to figure out--
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we need to integrate the
product of these two things,
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but I need to figure
out what dm actually is.
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So let me remind you also
what dm is and then we'll
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put it all together.
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dm, to remind you, is supposed
to be the density times dV.
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And the density in this
case was square root
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of x squared plus y squared,
that's just equal to r.
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So it's just r*dV.
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So there's a bunch
of little pieces
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and now I have to
put them together.
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So if I want to determine
the full attraction
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in the z-direction, because
the other two are 0,
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I just need to do
this triple integral.
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I'll figure out my
bounds momentarily.
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I'm going to leave a lot of room
to write my bounds, because I
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always run out of room.
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And then I just
need to integrate
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G m naught divided by rho
squared times z divided
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by rho times r dV.
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Actually I won't even
do the bounds yet
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because I have to
change everything
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into polar coordinates.
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So let's figure
out how to do that.
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What is z in polar
coordinates? z
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in polar coordinates we
know is rho cosine phi.
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So I'm going to
have a G m naught--
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I'm going to keep this
rho cubed right here--
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and then I'm going to
have a rho cosine phi.
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r is rho sine phi, so I'll
just put another rho, and then
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a sine phi.
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And then dV is rho squared
sine phi d rho d phi d theta.
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So what I'm going
to do here is--
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I have to add in a rho squared,
so I'm just going to cube this.
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And another sine phi, so
I'm going to square that.
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And then I'm going to have
a d rho d phi d theta.
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Hopefully that wasn't too scary.
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But the z is my rho cosine phi.
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The r is a single rho sine phi.
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And then dV includes a
rho squared sine phi,
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so I put a cubed here
and a squared here,
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and then d rho d phi d theta.
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So now I just have
to integrate, so I
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have to figure out my bounds.
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So we're integrating
first in rho.
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And rho, it's a sphere- a
half sphere of radius a.
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It's a solid half sphere.
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So I'm just integrating
from 0 up to a.
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And then phi-- because
it's a solid half sphere,
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phi is starting at 0
and it's going down
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until it hits the xy-plane.
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And so that's pi over 2.
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We know we start at 0, we go to
pi over 2 when we go a quarter
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turn, basically.
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When we go 90 degrees, right?
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So we go 0 to pi over
2, and then theta--
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once I've taken my rho and
I've gone up as far as I can,
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and I've gone down this way, I
have to rotate it all the way
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around-- and so theta is
actually going from 0 to 2*pi.
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All right.
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So this is my full
integral I have here.
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And now let's do
some simplifying
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before we integrate.
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Notice I have a rho cubed
in the denominator here
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and a rho cubed in
the numerator there.
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So when I integrate in rho,
I only have this single rho.
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When I integrate that rho,
everything else is fixed.
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So I integrate that rho, I
get a rho squared over 2,
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and I evaluate it
at 0 and a and so I
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get an a squared over 2,
as integrating this part.
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I'm going to pull
all that to the front
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and then write what I get next.
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Let me write what I get next.
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I'm going to have a G
m naught as a constant.
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I'm going to be multiplying
by a squared over 2
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from the rho integral.
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And then I have
2 more integrals.
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Integral from 0 to 2*pi.
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And the integral from 0 to
pi over 2 of cosine phi sine
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squared phi d phi d theta.
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So how do I integrate
this in phi?
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Looks like it's going to
be sine cubed phi over 3.
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Let me just double check.
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The derivative of that is 3
sine squared phi cosine phi.
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Yes.
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So, this is going to be
sine cubed phi over 3
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evaluated at 0 and pi over 2.
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This inner part is.
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And so let's see
what we get there.
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Sine of 0 is 0.
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Sine of pi over 2 is 1.
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So I just pick up a
1/3 from this integral.
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So now I'm going to
pull that out in front.
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So again, the thing
I've written down here
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is just the integral of this.
00:08:03.950 --> 00:08:07.997
So I just get a 1/3 out of that.
00:08:07.997 --> 00:08:08.830
Let me double check.
00:08:08.830 --> 00:08:10.521
Yep, sine of pi
over 2 is still 1.
00:08:10.521 --> 00:08:11.020
Good.
00:08:11.020 --> 00:08:12.490
So I get a 1/3 out of that.
00:08:12.490 --> 00:08:13.240
That's a constant.
00:08:13.240 --> 00:08:14.670
So I'll pull that to the front.
00:08:14.670 --> 00:08:16.128
And then notice
what I'm left with.
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I'm just integrating
0 to 2*pi of d theta.
00:08:19.070 --> 00:08:20.850
Well that's just 2*pi.
00:08:20.850 --> 00:08:22.915
So I've got a 1/3
from the inside one,
00:08:22.915 --> 00:08:25.310
times a 2*pi from
the outside one.
00:08:25.310 --> 00:08:30.750
So I just end up getting
2*pi times-- or 2*pi over 3,
00:08:30.750 --> 00:08:36.070
because there's my 1/3 also--
times G m naught a squared over
00:08:36.070 --> 00:08:37.000
2.
00:08:37.000 --> 00:08:41.300
And if I simplify
that, I just get
00:08:41.300 --> 00:08:46.290
G m naught a squared pi over 3.
00:08:46.290 --> 00:08:50.270
And so if I want to know
the force of this solid half
00:08:50.270 --> 00:08:54.160
sphere acting on the mass
m naught, where the mass is
00:08:54.160 --> 00:08:57.550
positioned at the center of the
base of the solid half sphere,
00:08:57.550 --> 00:08:59.540
it's going to be exactly
the following thing.
00:09:02.130 --> 00:09:04.920
By symmetry, the first
two components are 0
00:09:04.920 --> 00:09:11.440
and the last component is this
G m naught a squared pi over 3.
00:09:11.440 --> 00:09:14.290
And so that's my final solution.
00:09:14.290 --> 00:09:17.790
Let me just take us back and
remind us where we came from.
00:09:17.790 --> 00:09:22.590
So we wanted to find the force,
the gravitational attraction
00:09:22.590 --> 00:09:26.600
of this solid half sphere that
had a radius a, was centered
00:09:26.600 --> 00:09:29.070
at O, and had mass m
naught sitting at O.
00:09:29.070 --> 00:09:31.070
And so we were interested
in drawing the easiest
00:09:31.070 --> 00:09:31.800
picture possible.
00:09:31.800 --> 00:09:33.258
We're going to put
O at the origin,
00:09:33.258 --> 00:09:34.620
so m naught's at the origin.
00:09:34.620 --> 00:09:36.190
Then we have a solid
half sphere of radius a.
00:09:36.190 --> 00:09:38.231
So this is-- you know, if
I come all the way out,
00:09:38.231 --> 00:09:40.280
my radius length is a.
00:09:40.280 --> 00:09:43.770
And I gave you the
density was equal to r.
00:09:43.770 --> 00:09:46.220
So we know-- the force
we're interested in,
00:09:46.220 --> 00:09:48.620
F sub x, F sub y, F sub
z, from the picture alone,
00:09:48.620 --> 00:09:50.860
we see F sub x
and F sub y are 0.
00:09:50.860 --> 00:09:53.670
So we really only
need the z-component.
00:09:53.670 --> 00:09:55.570
So you know how to
find the magnitude
00:09:55.570 --> 00:09:58.626
and you know how to find the
direction of the vector field.
00:09:58.626 --> 00:10:00.625
The reason that you've
divided by rho here, just
00:10:00.625 --> 00:10:02.166
to remind you, is
that the direction,
00:10:02.166 --> 00:10:04.790
you want it to be a unit vector.
00:10:04.790 --> 00:10:06.680
And so this is what
makes it a unit vector.
00:10:06.680 --> 00:10:09.160
And so the direction part
that I'm interested in
00:10:09.160 --> 00:10:10.470
is the z-part.
00:10:10.470 --> 00:10:12.380
So it's z over rho.
00:10:12.380 --> 00:10:15.420
And so all I need to do
is take the z over rho,
00:10:15.420 --> 00:10:20.230
multiply it by the magnitude,
and replace the dm by what it
00:10:20.230 --> 00:10:21.910
is in terms of the volume form.
00:10:21.910 --> 00:10:24.910
And in terms of the volume
form, it's the density times dV
00:10:24.910 --> 00:10:27.240
and so in this case
it's just r*dV.
00:10:27.240 --> 00:10:29.490
So I made all those
substitutions.
00:10:29.490 --> 00:10:31.701
And then all I had to do
is do a lot of replacement.
00:10:31.701 --> 00:10:33.200
And that was kind
of the messy part,
00:10:33.200 --> 00:10:35.050
and then the
integral's fairly easy.
00:10:35.050 --> 00:10:38.690
So the messy part is z I
replaced by rho cosine phi.
00:10:38.690 --> 00:10:40.810
r I replaced by rho sine phi.
00:10:40.810 --> 00:10:45.264
And dV I replaced by rho squared
sine phi d rho d phi d theta.
00:10:45.264 --> 00:10:46.680
And then I just
have to figure out
00:10:46.680 --> 00:10:49.265
what the bounds are for the
half sphere, solid half sphere.
00:10:49.265 --> 00:10:50.140
And then I integrate.
00:10:50.140 --> 00:10:51.690
And the integration
is all ultimately
00:10:51.690 --> 00:10:54.110
one step at a time,
single variable.
00:10:54.110 --> 00:10:56.870
And they were all
fairly simple integrals.
00:10:56.870 --> 00:10:59.790
So in the end, we see that
your mass m naught sitting
00:10:59.790 --> 00:11:04.490
at the origin, the force exerted
on it by this solid half sphere
00:11:04.490 --> 00:11:08.960
is 0, 0 comma G times m naught
times a squared pi divided
00:11:08.960 --> 00:11:09.870
by 3.
00:11:09.870 --> 00:11:12.150
All right, that's
where I'll stop.