Video Transcript
𝐴𝐵𝐶𝐷 is a rectangle, where the
side 𝐴𝐵 equals 24 centimeters, and the side 𝐵𝐶 equals seven centimeters. Two forces, each of magnitude 43
newtons, are acting along 𝐵𝐴 and 𝐷𝐶, respectively. Determine the magnitude of each of
the two forces acting at points 𝐵 and 𝐷 and perpendicular to the line 𝐵𝐷 that
would make the whole system in equilibrium.
Let’s begin with a diagram of the
scenario. We have the rectangle 𝐴𝐵𝐶𝐷 with
side lengths 24 centimeters and seven centimeters. We have two forces of magnitude 43
newtons acting along 𝐵𝐴 and 𝐷𝐶. We have two forces, which we can
call 𝐹 one and 𝐹 two, acting at the points 𝐵 and 𝐷, and perpendicular to the
line 𝐵𝐷.
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 𝐹 one and 𝐹 two
are also antiparallel, for the system to remain in linear equilibrium, 𝐹 one and 𝐹
two must balance each other as well, and so they must be the same magnitude. Let’s call this magnitude 𝐹.
This system is also in rotational
equilibrium, so the couple comprising the forces 𝐹 one and 𝐹 two must be equal and
opposite to the couple comprising the two forces of 43 newtons. Recall that the magnitude 𝑀 of a
couple generated by two forces of magnitude 𝐹 acting perpendicularly to the ends of
a line of length 𝑑 is equal to 𝐹 times 𝑑. Looking at the line 𝐵𝐶, we have
two forces of 43 newtons acting from the points 𝐵 and 𝐶 and perpendicularly to the
line 𝐵𝐶, which has length seven centimeters.
Therefore, the magnitude 𝑀 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 𝐹 one and 𝐹 two. This second couple, 𝑀 two, is also
given by the magnitude of the unknown forces 𝐹 multiplied by the length of the line
between their points of action, 𝐵𝐷.
Rearranging this equation for 𝐹
gives us 301 divided by the length of the diagonal line 𝐵𝐷, 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.
