7.1:
Work
Work is said to be done when energy is transferred from one entity to another via the application of force. For example, when a force F is applied on a box and it moves through a displacement ds, work is done against the force of friction.
The increment in the work done during the process is equal to the dot product of the force and displacement vectors.
Only the component of force applied parallel to the displacement of the object contributes towards the work done.
When an object is moved from position A to B, the total work done by the force is the integral of the force with respect to the displacement along the path of the displacement.
For a force that is constant in both magnitude and direction, the integral depends only on the endpoints, and hence the work done is independent of the path taken.
When a variable force is acting on an object, like expansion or compression of a spring, the spring force is expressed as a function of distance. The work done by the spring force along the displacement from initial position to final position is given by this equation, where k is the spring constant.
7.1:
Work
Work is done when energy is transferred from one object to another. In other words, work is when a force acts on something that undergoes a displacement from one position to another. Forces can vary as a function of position, and displacements can be along various paths between two points. The increment of work (dW) done by a force acting through an infinitesimal displacement can be defined as the dot product of force (
) and displacement (
) vectors.
The dot product can be expressed in terms of the magnitudes of the vectors and the cosine of the angle between them, as it is easier to define the dot product in words as an expression of magnitudes and angles. It can also be expressed in terms of various components, as introduced in the vectors lesson.
From vector properties, it does not matter if you take the component of the force parallel to the displacement, or the component of the displacement parallel to the force—the result is the same either way. The units of work are units of force multiplied by units of length; this is Newtons multiplied by meters (N·m) in the SI system. In the English system, the unit of force is the pound (lb), and the unit of distance is the foot (ft), so the unit of work is the foot-pound (ft·lb).
The work done by a force that is constant in magnitude and direction is the simplest to calculate. In general, forces can vary in magnitude and direction at points in space, and the paths between two points can be curved. The infinitesimal work done by a variable force can be expressed in terms of the components of the force and the displacement along the path. In these cases, the components of the force are functions of position along the path, and the displacements depend on the equations of the path. However, the physical concept of work is straightforward: calculate the work for tiny displacements and add them up.
This text is adapted from Openstax, University Physics Volume 1, Section 7.1: Work.