 In engineering, you will hear the term moment of a force, or simply just moment, used a lot. But what is it exactly? We are well aware that a force can push or pull an object into motion, translating in the direction of that force. This is also captured in Newton's second law, force equals mass times acceleration, where in vector form the force and acceleration vectors will have the same direction. But there is more to motion than just translation. Rotation of a body about an axis is another type of motion that crops up a lot in our daily lives. You may notice it in the wheel of your bike or car, or at the home of that friend who really is into vinyl records. We use rotation to generate power on the ground and to provide power to the machines we build to fly. Rotations are all around us, but what causes rotation? If we think about an aircraft represented by its center of gravity, we have seen that the lift and the weight are equal and opposite when the aircraft is flying straight and level. That is when the acceleration is zero and it is in static equilibrium. If the lift is increased, the aircraft will begin to accelerate in a vertical direction and gain altitude. But we know an aircraft can also pitch up and down. It achieves this through control surfaces such as the elevator, which generates a force that is not in line with the center of gravity. And this offset force causes the aircraft to rotate about its center of gravity. This tendency for the rotation is proportional to the magnitude of the force and the perpendicular distance d between the force and the axis of rotation. We call this tendency for rotation the moment of a force, or simply moment. As mentioned earlier, this moment is quantified by the scalar product of the force f and the perpendicular distance d, also known as the moment arm between the force and the axis of rotation. The moment of a force is itself a vector. So what would be the direction of this vector? It is easiest to first think about it in terms of the rotation that a moment would tend to impart on an object. Take this top, for instance. If a force was applied on the edge of the top as shown here, it would want to rotate around its spinning axis in the direction shown here. This is intuitive for a top, but can we formulate a method for resolving the direction for a general force system? We certainly can with a little help from our right hand. Using what is known as the right hand rule, we can determine the direction of any moment of a force. To apply the right hand rule, all you need to do is point your fingers from the axis or the point where the moment is acting to the force that is causing the moment along the moment arm d. Then you bend your fingers of your right hand to point in the direction of the force. You may need to flip your hand over in order to do this. The direction your fingers curl is the direction of rotation that the moment is tending to induce on a particular body. But remember, we said that a moment is a vector and the vector will point along the axis of rotation. So to find this direction, give your right hand a thumbs up and you will have your thumb pointing in the direction of the moment vector. We tend to use a double headed notation for the moment as shown here, as it is a little bit more clear than trying to show a rotation direction as a curved arrow in our problems, particularly in 3D problems. Summarizing what we have seen thus far, we understand that a moment is a measure of the tendency of a force to cause rotation about a given point or axis. It can be quantified as the scalar product of the force f and the perpendicular distance d between the point of rotation and the force. Also, the direction of the moment can be visualized using the right hand rule. But this is all in scalar formulation. Isn't there a vector formulation as well? Indeed there is, and it uses something you have recently learned in calculus, the cross product. The vector formulation of the moment of a force is m is equal to r cross f, where the cross indicates the vector cross product. One of the very convenient things about the vector formulation of the moment is that the position vector r does not need to be the perpendicular distance between the rotation axis and the force. In fact, it can be any position vector between the axis of rotation and the line of action of the force, as long as it is in the direction from the axis to the force as shown here. Comparing this vector formulation to the scalar formulation of a moment, you may observe something a bit odd. In scalar formulation, the moment is given as the force f multiplied by distance d, while for the vector formulation, it is the position vector r cross force f. For scalar multiplication, the order of these terms does not matter. f multiplied by d is equal to d multiplied by f. But this is not true for the cross product. r cross f does not equal f cross r. The resulting vector from these two cross products will have the same magnitude, but they will have different directions. You can visualize this with the right-hand rule we described earlier by swapping the role of f and r. I have one last thing to show you to help you in calculating moments in your engineering problems. It is known as the principle of moments or Vérignon's theorem after the French mathematician that proved it. The principle of moments relates a moment of a force to the moments of that force decomposed into different components. So if we consider the force f shown here and decompose it into components f1 and f2, we can modify our moment equation to give us m is equal to r cross in brackets f1 plus f2. This can then be expanded to r cross f1 plus r cross f2. So the moment of a force is equal to the sum of the individual moments caused by the components of that force. That was certainly a lot to take in, but you will find that as you practice calculating moments and solving all forms of problems throughout your engineering degree, a lot of these concepts will become second nature to you. Good luck in the remaining learning activities in the rest of this module.