### Video Transcript

A uniform hemispherical shell of
mass 21 kilograms rests on a smooth horizontal plane. A particle of mass 14 kilograms is
placed on the rim of this shell, causing it to tilt so that the plane of the rim is
at an angle of ๐ผ to the horizontal when the system is in equilibrium. Find the value of tan of ๐ผ.

Letโs begin by drawing a free body
diagram. Here is our uniform hemispherical
shell. Weโre going to let it have a radius
of ๐ units. A particle of mass 14 kilograms
rests on the rim of this shell, causing it to tilt. We know that the downwards force of
the weight of this particle is mass times gravity, so 14๐. And we also know that the plane of
the rim is at an angle of ๐ผ to the horizontal.

But what do we do about the
downwards force of the weight of our uniform hemispherical shell? Well, because itโs uniform, we can
quote a formula to help us find the center of mass. The center of mass of our uniform
hemispherical shell lies on the axis of symmetry at a distance of ๐ over two from
the point ๐. Well, here is the axis of
symmetry. And so our downwards force of 21๐
must act here.

Now that we have the diagram, weโre
going to see what it means for the system to be in equilibrium. Firstly, we know that for a body to
be in equilibrium, the sum of all forces acting on that body must be equal to zero
and the sum of all moments acting on the body must be equal to zero, where the
moment, which is the turning effect of a force, is calculated by multiplying the
magnitude of that force by the perpendicular distance of the line of action of that
moment from the point about which the object is trying to turn.

And so letโs imagine weโre going to
take moments about ๐. We have a force of 21๐ and a force
of 14๐. We need to calculate the
perpendicular distance from ๐ to the line of action of these forces. Letโs begin with the 14๐
force. We can add a right-angled triangle
in here. Alternate angles are equal, so the
included angle is ๐ผ. And letโs call the side that weโre
trying to find, which is the perpendicular distance from ๐ to that downward force
14๐, ๐ฅ. Relative to the included angle, we
want to find the length of the adjacent side. And weโve actually defined the
hypotenuse to be equal to ๐.

So letโs link these by using the
cosine ratio such that cos of ๐ผ is ๐ฅ over ๐. Weโll find an expression for ๐ฅ by
multiplying through by ๐. And so weโll find that ๐ฅ is equal
to ๐ cos ๐ผ. We add that to the diagram. And now we consider the 21๐
force. Here is our triangle. Now the included angle down here is
๐ผ. We can convince ourselves that this
is true since we know that the edge of the hemisphere and the line of action of the
center of the mass to ๐ must meet at 90 degrees.

Angles on a straight line sum to
180. So this angle here is 90 minus
๐ผ. And then if we subtract 90 minus ๐ผ
and 90 from 180 degrees, thatโs the interior angle sum of a triangle, we get our
included angle to be ๐ผ. This time then, weโre interested in
the opposite side and the hypotenuse of this triangle. So weโre going to use the sine
ratio. This time, sin of ๐ผ is ๐ฆ over ๐
over two. And we can solve for ๐ฆ by
multiplying by ๐ over two. So ๐ฆ is ๐ over two sin ๐ผ. Letโs add this to the diagram. And weโre ready to take
moments.

Now when we take moments about a
point, we do need to define a positive direction. Letโs say that the counterclockwise
direction here is positive. Thinking about our 21๐ force, we
multiply force by perpendicular distance. Thatโs 21๐ times ๐ over two sin
๐ผ. This force is trying to turn the
object in a counterclockwise direction. And so its moment is going to be
positive. The other force, the 14๐ force, is
trying to turn the object in a clockwise direction. And so its moment is going to be
negative.

Force times perpendicular distance
is 14๐ times ๐ cos ๐ผ. And we know that the body is in
equilibrium. So letโs set this equal to
zero. Remember, weโre trying to find the
value of tan of ๐ผ. And we have two variables at the
moment. We have ๐ and ๐ผ. Of course, ๐ is the radius of our
hemisphere. And so it cannot be equal to
zero. This means we can divide our entire
equation by ๐. Similarly, ๐ is the acceleration
due to gravity. Itโs about 9.8. And so we can divide through by
๐.

While weโre here, we might also
spot that we can divide through by seven. 21 divided by seven is three, and
14 divided by seven is two. And so our equation becomes three
over two sin ๐ผ minus two cos ๐ผ equals zero. So we have an equation purely in
terms of ๐ผ. But how do we link sin ๐ผ and cos
๐ผ to tan ๐ผ? Well, we know the trigonometric
identity tan ๐ผ is equal to sin ๐ผ over cos ๐ผ. And so we need to manipulate our
equation. Weโre going to do this by adding
cos ๐ผ to both sides.

So we get three over two sin ๐ผ
equals two cos ๐ผ. Now if we divide through by cos ๐ผ,
on the left-hand side weโre going to have sin ๐ผ over cos ๐ผ. So three over two sin ๐ผ over cos
๐ผ equals two. And so we replace sin ๐ผ over cos
๐ผ with tan ๐ผ.

Weโre nearly finished. To find the value of tan of ๐ผ, we
just need to divide through by three over two. tan of ๐ผ then is two divided by
three over two, which is the same as two times two over three, which is
four-thirds. Weโre, therefore, able to say then
that the value of tan ๐ผ under these circumstances is four-thirds.