### Video Transcript

π΄π΅ is a rod having a length of 90
centimeters and a negligible weight. It is suspended horizontally by a
pin at its midpoint. Two forces each of magnitude 7.5
newtons are acting at its ends, as shown in the figure. It is also pulled by a string whose
tension is 25 newtons in a direction making an angle of 30 degrees with the rod from
point πΆ. If a force πΉ is acting on the rod
at point π· so that the rod is in a horizontal equilibrium position, find the
magnitude of πΉ, its direction π, and the length πΆπ·.

As the rod is in equilibrium, we
know that the sum of the forces in the π₯- or horizontal direction equals zero. Likewise, the sum of the forces in
the π¦- or vertical direction also equals zero. It will also be true that the sum
of the moments about any point in the system will equal zero. We will take the positive
directions to be to the right, vertically upward, and counterclockwise. We are told that the rod is of
length 90 centimeters. If we let the point π be the
center or midpoint of the rod, we know that π΄π is equal to ππ΅, which is equal to
45 centimeters as this is a half of 90.

We will now clear some space so we
can calculate the three unknowns required. The 25-newton force and πΉ-newton
force are not acting horizontally or vertically. This means that we need to find the
horizontal and vertical components before we resolve in these directions. We can do this using our knowledge
of right-angled trigonometry, where the sine of angle π is equal to the opposite
over the hypotenuse and the cosine of angle π is equal to the adjacent over the
hypotenuse.

If we consider the force at point
πΆ, we will let the horizontal component be equal to π₯ and the vertical component
equal to π¦. This means that the sin of 30
degrees will be equal to π¦ over 25 and the cos of 30 degrees is equal to π₯ over
25. We can multiply both sides of these
equations by 25. This means that π¦ is equal to 25
multiplied by the sin of 30 degrees, and π₯ is equal to 25 multiplied by the cos of
30 degrees. The sin of 30 degrees is equal to
one-half, and the cos of 30 degrees is equal to root three over two. This means that we have a vertical
component equal to 25 over two or 12.5 newtons and a horizontal component equal to
25 root three over two newtons.

We can repeat this for the
πΉ-newton force at point π·. This time, we have a horizontal
component acting to the left equal to πΉ cos π and a vertical component acting
downward equal to πΉ sin π. We can now resolve in both the
horizontal and vertical directions. The forces acting in a horizontal
direction are 25 root three over two and πΉ cos π. As the sum of the forces is equal
to zero and the πΉ cos π is moving to the left, we have 25 root three over two
minus πΉ cos π is equal to zero. 25 over two is equal to 12.5, so
this simplifies to 12.5 root three minus πΉ cos π is equal to zero. We can then add πΉ cos π to both
sides, giving us an expression for this equal to 12.5 root three. We will call this equation one.

There are four forces acting in a
vertical direction. From left to right, these are
negative 7.5, positive 12.5, negative πΉ sin π, and positive 7.5. We know that these must also sum to
equal zero. The two 7.5-newton forces are an
example of a coplanar couple. They have the same magnitude but
act in the opposite direction. This means that when resolving
vertically, these forces will cancel as negative 7.5 plus 7.5 is equal to zero. Adding πΉ sin π to both sides of
our equation gives us 12.5 is equal to πΉ sin π. We will call this equation two.

We now have a pair of simultaneous
equations with two unknowns, πΉ and π. Dividing equation two by equation
one, we get 12.5 over 12.5 root three is equal to πΉ sin π over πΉ cos π. We can divide the numerator and
denominator of the left-hand side by 12.5, giving us one over root three. We can divide the numerator and
denominator of the right-hand side by πΉ. This leaves us with sin π over cos
π, which we know is equal to tan π. We can then take the inverse
tangent of both sides of this equation such that π is equal to the inverse tan of
one over root three. This gives us a value of π equal
to 30 degrees. We have calculated the direction of
the force πΉ.

Next, we need to substitute π
equals 30 degrees into equation one or equation two. If we substitute into equation two,
we have 12.5 is equal to πΉ sin 30. The sin of 30 degrees is equal to
one-half. So, 12.5 is equal to πΉ multiplied
by one-half. We can then divide both sides of
this equation by one-half or multiply them both by two to give us a value of πΉ
equal to 25 newtons. The magnitude of force πΉ is 25
newtons.

We will now clear some space and
use moments to calculate the length of πΆπ·. Before doing this, however, it is
worth noting that the force πΉ and the 25-newton force are a coplanar couple. This is due to the fact theyβre of
equal magnitude and act in the opposite direction. The two forces are parallel to one
another. This means that the system in this
question consists of two pairs of coplanar couples. We will take moments about the
midpoint π. But before doing this, we will tidy
up our diagram so we have the relevant forces and distances.

As we have two pairs of coplanar
couples, the distance πΆπ must be equal to the distance ππ·. We will call this distance π₯
centimeters. We know that the moment of a force
is equal to its magnitude multiplied by the perpendicular distance. The force at π΄ is acting in a
counterclockwise direction. Therefore, the moment is equal to
7.5 multiplied by 45. The force at πΆ is acting in a
clockwise direction; therefore, this is equal to negative 12.5π₯. The vertical force at π· has the
same magnitude. This is also acting in the
clockwise direction and is equal to negative 12.5π₯. Finally, the moment of the force at
π΅ is acting in a counterclockwise direction and once again is equal to 7.5
multiplied by 45. We know that the sum of these four
values equals zero.

This simplifies to 337.5 minus
12.5π₯ minus 12.5π₯ plus 337.5 is equal to zero. Collecting like terms, we have 675
minus 25π₯ is equal to zero. Adding 25π₯ to both sides, we have
25π₯ is equal to 675. And then dividing both sides by 25
gives us π₯ is equal to 27. This means that the lengths of πΆπ
and ππ· are equal to 27 centimeters. 27 plus 27 is equal to 54. Therefore, the length πΆπ· is 54
centimeters. We now have our three answers. The magnitude of πΉ is 25 newtons,
the direction π is 30 degrees, and πΆπ· is 54 centimeters.