### Video Transcript

π΄π΅πΆπ· is a rectangle, where the
side π΄π΅ equals 24 centimeters, and the side π΅πΆ equals seven centimeters. Two forces, each of magnitude 43
newtons, are acting along π΅π΄ and π·πΆ, respectively. Determine the magnitude of each of
the two forces acting at points π΅ and π· and perpendicular to the line π΅π· that
would make the whole system in equilibrium.

Letβs begin with a diagram of the
scenario. We have the rectangle π΄π΅πΆπ· with
side lengths 24 centimeters and seven centimeters. We have two forces of magnitude 43
newtons acting along π΅π΄ and π·πΆ. We have two forces, which we can
call πΉ one and πΉ two, acting at the points π΅ and π·, and perpendicular to the
line π΅π·.

Without these two forces, the
system is already in linear equilibrium, since the two forces of 43 newtons are
antiparallel and will balance each other. Therefore, since πΉ one and πΉ two
are also antiparallel, for the system to remain in linear equilibrium, πΉ one and πΉ
two must balance each other as well, and so they must be the same magnitude. Letβs call this magnitude πΉ.

This system is also in rotational
equilibrium, so the couple comprising the forces πΉ one and πΉ two must be equal and
opposite to the couple comprising the two forces of 43 newtons. Recall that the magnitude π of a
couple generated by two forces of magnitude πΉ acting perpendicularly to the ends of
a line of length π is equal to πΉ times π. Looking at the line π΅πΆ, we have
two forces of 43 newtons acting from the points π΅ and πΆ and perpendicularly to the
line π΅πΆ, which has length seven centimeters.

Therefore, the magnitude π one of
the couple comprising these two forces is 43 times seven, which is equal to 301
newton-centimeters. This couple will have equal
magnitude to the couple comprising the forces πΉ one and πΉ two. This second couple, π two, is also
given by the magnitude of the unknown forces πΉ multiplied by the length of the line
between their points of action, π΅π·.

Rearranging this equation for πΉ
gives us 301 divided by the length of the diagonal line π΅π·, which is given via the
Pythagorean theorem by the square root of the squares of the sides of the rectangle,
24 and seven. Performing this calculation gives
us the magnitude of both the unknown forces, 12.04 exactly, and the unit is
newtons.