𝐴𝐵𝐶𝐷 is a square of side length eight centimeters, where two forces of magnitudes 21 newtons are acting at 𝐵 and 𝐷, respectively, and their lines of action are in the direction of 𝐴𝐶 and 𝐶𝐴, respectively. Determine the magnitude of the moment of the couple.
We will begin by sketching a diagram to model the situation. We are told that 𝐴𝐵𝐶𝐷 is a square of side length eight centimeters. We have two forces acting at point 𝐵 and 𝐷. They are of magnitude 21 newtons and act in the direction of 𝐴𝐶 and 𝐶𝐴, respectively. Adding these forces onto our diagram, we notice that they act in the opposite direction and are of equal magnitude. In fact, they form a couple. They are a pair of parallel forces with equal magnitudes and opposite direction which do not lie on the same line of action.
We are asked to calculate the magnitude of the moment of this couple. And this is equal to 𝐹𝑑 sin 𝜃, where 𝐹 is the magnitude of both forces in the couple, lowercase 𝑑 is the distance between the points of action of both forces, and 𝜃 is the angle between either force and the line segment that joins them. In this question, we are told that 𝐹 is 21 newtons. We can see from our diagram that the angle 𝜃 is 90 degrees. We will use our knowledge of the Pythagorean theorem to calculate the length 𝑑.
As this is the hypotenuse of our triangle, 𝑑 squared is equal to eight squared plus eight squared. We know that eight squared is equal to 64. 𝑑 squared is therefore equal to 128. Taking the square root of both sides and noting that 𝑑 must be positive. We have 𝑑 is equal to the square root of 128. Using our laws of radicals or surds, we can rewrite this as root 64 multiplied by root two. And since root 64 is equal to eight 𝑑 is equal to eight root two.
The magnitude of the moment of the couple is therefore equal to 21 multiplied by eight root two multiplied by sin of 90 degrees. And as the sin of 90 degrees is equal to one, this simplifies to 21 multiplied by eight root two. 21 multiplied by eight is 168. We can therefore conclude that the magnitude of the moment of the couple is 168 root two newton-centimeters.