### Video Transcript

A body of weight 90 kilogram-weight
is placed on a smooth plane that is inclined at 30 degrees to the horizontal. If the body is held in equilibrium
by means of a force π
that acts at an angle of 30 degrees above the plane,
determine the magnitudes of π
and π, where π is the reaction of the plane on the
body.

We will begin by sketching a
diagram that models the situation. A body of weight 90 kilogram-weight
is placed on a smooth plane. We are told that the plane is
inclined at an angle of 30 degrees. The body is held in equilibrium by
a force π
. And this acts at an angle 30
degrees above the plane. We are asked to calculate the
magnitude of this force π
together with π, which is the reaction of the plane on
the body. The reaction force will act in the
direction perpendicular to the plane.

In this question, we have three
forces acting at a point. Since the body is in equilibrium,
we can use Lamiβs theorem to help solve it. This states that when three forces
acting on a point are in equilibrium, then each force is proportional to the sine of
the angle between the other two forces. If we have three forces π΄, π΅, and
πΆ, then π΄ over sin πΌ is equal to π΅ over sin π½, which is equal to πΆ over sin
πΎ, where πΌ is the angle between forces π΅ and πΆ, π½ is the angle between the
forces π΄ and πΆ, and πΎ is the angle between the forces π΄ and π΅.

In our diagram, we will begin by
calculating the angles between the three forces. The angle between the force π
and
the reaction force is 60 degrees, as 90 minus 30 is equal to 60. This is the angle that is opposite
the weight force. The angle between the reaction
force and the weight force is 150 degrees, as this is 90 degrees plus 60
degrees. This angle is opposite the force
π
. Finally, the angle between the
weight force and the force π
is also 150 degrees, as angles at a point must sum to
360 degrees. This angle is opposite the reaction
force π.

We can now substitute these values
into Lamiβs theorem. We have π
over the sin of 150
degrees is equal to π over the sin of 150 degrees, which is equal to 90 over sin 60
degrees. The sin of 150 degrees is
one-half. And since dividing by a half is the
same as multiplying by two, the first two parts of our equation simplify to two π
and two π. The sin of 60 degrees is equal to
root three over two. 90 divided by root three over two
is 180 over root three. We can rationalize this by
multiplying the numerator and denominator by root three. This gives us 180 root three over
three, which in turn simplifies to 60 root three. Two π
and two π are both equal to
60 root three. Dividing through by two, we see
that π
is equal to π, which is equal to 30 root three.

We can therefore conclude that the
force π
is equal to 30 root three kilogram-weight and the reaction force π is also
equal to 30 root three kilogram-weight.