A Moment Is the Tendency to Rotate, M = F times Perpendicular Distance
Why is a door handle mounted on the side farthest from the hinge? Why do you reach for a longer wrench to loosen a stuck bolt? Because the same force turns things better the farther it acts from the axis of rotation. This tendency of a force to rotate a body is called the moment, also called torque. The size of the moment is the force times the perpendicular distance from the axis to the force's line of action. M = F d. Force alone is not enough; where and how you apply it matters just as much. Start by growing the force at the end of the wrench and feel how hard the bolt is turned.
Here we are loosening a bolt with a wrench. The bolt is the axis of rotation, and you apply a force F at the end of the wrench. Grow the force. The larger it is, the larger the tendency to turn the bolt, the moment, shown as a curved arrow. The key point is that this force does not slide the bolt sideways; it rotates it. The same force produces a turning effect thanks to where it acts. If force is the ability to push something straight, the moment is the ability to rotate it. They are different quantities with different units: force in N, moment in N·m. In the next scene we put a formula to this size.
The size of the moment is surprisingly simple. Take the force F and multiply by the distance d from the axis to the force, and you are done. M = F d. For now, take the force as perpendicular to the distance direction. Use the two sliders to change F and d. Grow either one and M grows the same way. Double the force or double the distance, and the turning effect doubles all the same. So even a weak force can make a strong rotation if it uses a long arm. This is exactly the principle of the lever. It helps to think of M as an area: the area of a rectangle with base d and height F is the moment.
But if the force is slanted, how do we measure the distance? The key is the perpendicular distance. The distance dropped straight from the axis to the force's line of action is called the moment arm. M = F times this moment arm. Turn the angle of the force. The farther the line of action runs from the axis, the longer the moment arm and the larger M; and when the line of action passes exactly through the axis, the moment arm is zero, so no matter how hard you push, M is zero. It is like pushing a door toward its hinge: it will not open. This is also why moving the force anywhere along its line of action leaves the moment unchanged: the perpendicular distance to the line does not change.
A moment has a direction, a sign. In a plane there are only two senses of rotation: counterclockwise and clockwise. By convention we take counterclockwise as positive and clockwise as negative. Turn the direction of the force. When the force tends to rotate the body counterclockwise, M is positive and green; when clockwise, it is negative and red. The instant the line of action passes through the axis, the moment goes through zero and the sign flips. In components, M = x Fy minus y Fx gives the sign in one shot, the z-component of the cross product of the position vector with the force. Once this sign convention is fixed, adding several moments is just a matter of attaching signs and summing.
Now let us add moments. A seesaw makes it easy. About the pivot, the weight on the left tends to turn it counterclockwise, the weight on the right clockwise. The net turning effect is ΣM, the two moments added with their signs. Change the force on the right. If ΣM is not zero the seesaw tips that way, and when it is exactly zero it balances level. This is why a light person seated far from the pivot can balance a heavy one: F may be small, but with a large d the moment M comes out the same. So ΣM = 0 is the equilibrium condition for rotation, and it is half the heart of the rigid-body equilibrium we study next.