moment of a force explained
Why does a door handle sit farthest from the hinge. Why reach for a longer wrench to free a tight bolt. The same force turns an object more easily the farther it acts from the axis. This tendency of a force to rotate an object is the moment, or torque. Its size is the force times the perpendicular distance from the axis to the line of action, M = F d, so the force alone is not enough and where it is applied matters too. Start by raising the force at the end of the wrench and watching how hard the bolt is turned.
A wrench frees a bolt. The bolt is the axis and a force is applied at the end of the wrench, and raising the force grows the moment, shown as a curved arrow. The point is that this force rotates the bolt rather than translating it, and the place it acts creates the turning effect. If force is the ability to translate, moment is the ability to rotate, and they are different quantities, so the unit of force is N and the unit of moment is N·m. The next step writes this size as an equation.
The size of a moment is simple. Multiply the force by the distance from the axis to the force and that is M = F d, taking the force perpendicular to the distance for now. Change the force and the distance and raising either one grows the moment the same way, and doubling the force or doubling the distance doubles the turning effect the same. A weak force on a long arm can still produce a strong rotation, and that is the principle of the lever. Think of the moment as an area, the rectangle with width d and height equal to the force.
When the force is slanted, how should the distance be measured. The key is the perpendicular distance, the moment arm, dropped perpendicular from the axis to the line of action. M = F times this arm, so turning the angle of the force lengthens the arm as the line of action moves away from the axis and the moment grows. When the line of action passes exactly through the axis the arm is 0 and however hard you push the moment is 0, the same story as pushing a door toward its hinge. Moving the force anywhere along its line of action leaves the moment unchanged, because the perpendicular distance to the line does not change.
A moment has a direction, a sign. In the plane rotation is only counterclockwise or clockwise, and the agreement is counterclockwise positive and clockwise negative. Turn the direction of the force and the moment is positive and green when the force tries to turn the object counterclockwise, negative and red when it tries to turn it clockwise. The moment passes through 0 and flips sign the instant the line of action crosses the axis. In components M = x Fy minus y Fx gives the sign in one step, the depth component of the cross product of position and force. With this agreement, several moments are added with their signs.
Now add the moments. A seesaw makes it easy: about the fulcrum the left weight tries to turn counterclockwise and the right weight clockwise, and the net turning effect is the two moments added with their signs. Change the right force and the seesaw tips that way when the sum is not 0, and it balances level when the sum is exactly 0. A light person balances a heavy one by sitting far from the fulcrum because a small force with a large distance can make the same moment. ΣM = 0 is the equilibrium condition for rotation, half of the rigid-body equilibrium that comes next.