## Quick Reference

(in mechanics)

A means of measuring the turning effect of a force about a point. For a system of coplanar forces, the moment of one of the forces **F** about any point *A* in the plane can be defined as the product of the magnitude of **F** and the distance from *A* to the line of action of **F**, and is considered to be acting either clockwise or anticlockwise. For example, suppose that forces with magnitudes *F*_{1} and *F*_{2} act at *B* and *C*, as shown in the figure. The moment of the first force about *A* is *F*_{1}*d*_{1} clockwise, and the moment of the second force about *A* is *F*_{2}*d*_{2} anticlockwise. The principle of moments considers when a system of coplanar forces produces a state of equilibrium.

However, a better approach to measuring the turning effect is to define the moment of a force about a point as a vector, as follows. The moment of the force **F**, acting at a point *B*, about the point *A* is the vector (**r**_{B}−**r**_{A})×**F**, where this involves a vector product. The use of vectors not only eliminates the need to distinguish between clockwise and anticlockwise directions, but facilitates the measuring of the turning effects of non-coplanar forces acting on a 3-dimensional body.

Similarly, for a particle *P* with position vector **r** and linear momentum **p**, the moment of the linear momentum of *P* about the point *A* is the vector (**r**−**r**_{A})×**p**. This is the angular momentum of the particle *P* about the point *A*.

Suppose that a couple consists of a force **F** acting at *B* and a force −**F** acting at *C*. Let **r**_{B} and **r**_{C} be the position vectors of *B* and *C*, and let **r**_{A} be the position vector of the point *A*. The moment of the couple about *A* is equal to (**r**_{B}−**r**_{A})×**F**+(**r**_{C}−**r**_{A})×(−**F**)=(**r**_{B}−**r**_{C})×**F**, which is independent of the position of *A*.

*Subjects:*
Mathematics.

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