Q1. What is a concurrent force system?

A scalar quantity is measured by its magnitude. For example, density and mass are two scalar quantities. In contrast, a vector quantity is defined in terms of both magnitude and direction. For example, force and moment are two vector quantities.

It is possible to determine the individual direction cosines if we know the magnitude of the vector and its individual components. For example, if the component of vector along the x axis is given by Rx, then the corresponding direction cosine can be found as

where R is the magnitude of vector . The other two direction cosines can be found in a similar way.

Each component represents the projection of the vector along a particular axis. For example, a three-dimensional vector can be represented in terms of its individual components in the Cartesian coordinate system as

where Rx, Ry and Rz are obtained as

R is the magnitude of and θx, θy, θz represent the angles between and x, y, z axes, respectively as shown in the figure.

The magnitude of a vector can be determined as

and its direction can be expressed in terms of the angles between the vector and x,y,z axes as

A vector whose magnitude is unity or one is called a unit vector. It is also important to note that a unit vector is a dimensionless quantity. It carries with it no units of measurement. In this module, we use to represent a unit vector.

with magnitude and direction cosines determined as

Yes, we can represent a force vector by the product of its magnitude and a unit vector that is in the same direction as the force. Mathematically, this relationship is expressed as

Notice that we can use a similar representation for any other vector quantity as well.

No! The polygon rule can only be used for finding the resultant of forces that are coplanar and concurrent. That means all the forces must be in the same plane and must pass through the same point in that plane. Furthermore, the triangle rule and parallelogram law can only be used to find the resultant of two concurrent forces. If a system contains more than two concurrent forces, then the parallelogram law should be applied to two forces at a time.

A particle is said to be in the state of equilibrium when the resultant of all the forces acting on it is zero. For a general three-dimensional force system, we can express equilibrium in terms of the x,y,z components of forces as

It is also important to realize that a particle can never be in the state of equilibrium with only a single force acting on it.

We use free body diagrams to identify the known as well as unknown forces acting on a particle. Then with the help of equilibrium equations we solve for the unknown forces.

It is also important to realize that a particle can never be in the state of equilibrium with only a single force acting on it.

The most important thing about a direction angle is that it is the angle between the vector and the positive axis of interest. Whether we measure this angle in the clockwise or counter clockwise direction is not important. We should also keep in mind that a direction angle cannot exceed 180 degrees, as shown in the figure below.