4.6 : Resultant Moment: Vector Formulation

When a force is applied to an object, the tendency of the object to rotate about a point is known as its moment. If multiple forces are acting on an object, the sum of moments of all the forces acting on a body can be expressed as the resultant moment of the system. The resultant moment can be considered a vector quantity that can be added and subtracted like any other vector.

The resultant moment of a system of forces can be calculated through vector formulation. For example, if we consider two forces, F₁ and F₂, acting on a pole at points A and B, respectively, the resultant moment of these forces can be expressed as

M_R = (r_A x F₁)+ (r_B x F₂).

where r_A and r_B are the position vectors from a fixed point to the points A and B, respectively. The product r x F is known as the moment of the force, and they can be expressed in determinant form. Considering the cross-product of the position vector with the force vector takes into account the direction of the force and the point about which it causes rotational motion.

This vector formulation of the resultant moment has several practical applications. For example, engineers use it to calculate the torque on machine parts, such as gears, pulleys, and cranks. It is also widely used in sports sciences to calculate the torques on joints during different types of physical activity.