In the figure, the rod is fixed by a freely rotating hinge at 𝐴. If a vertical force of magnitude 120 newtons acts downwards as shown at end 𝐵, find the magnitude of the moment of the force about 𝐴.
Let’s begin by recalling what we mean when we talk about the moment of a force about a point. The moment of a force is a measure of its tendency to cause a body to rotate about a specific point. We calculate the moment by multiplying the magnitude of the force by the perpendicular distance of the line of action of the force from the point at about which it’s trying to pivot.
Now, in fact, the 120-newton force is acting vertically downward here. So we’re going to calculate the component of this force, which is perpendicular to the line segment 𝐴𝐵. In other words, if we add a right-angled triangle where the component of the force that’s perpendicular to the line segment 𝐴𝐵 forms an angle of 30 degrees to the downwards force 120 newtons, we can use right triangle trigonometry. We’re trying to find the adjacent, and we know the hypotenuse.
In this case then, we use the cosine ratio, where cos of 𝜃 is adjacent over hypotenuse. So here, cos of 30 degrees is equal to 𝑥 over 120, where we’ll define 𝑥 to be the component of the force that’s perpendicular to the rod. To solve for 𝑥, we multiply both sides of this equation by 120. So 𝑥 is 120 times cos 30. But of course, cos of 30 degrees is root three over two. So this becomes 60 root three newtons.
We’re now ready to calculate the moment of this force about point 𝐴. We generally define the counterclockwise direction to be positive. But since we’re trying to find the magnitude of the moment of the force and this force is trying to turn the rod in a clockwise direction, we’ll define the positive direction to be clockwise. Since 120 centimeters is equal to 1.2 meters, the moment is the force times the distance. It’s 60 root three times 1.2. That’s 72 root three, and our units are newton meters. So the magnitude of the moment of the force about 𝐴 is 72 root three newton meters.