Work (physics) - Mathematical Calculation

Mathematical Calculation

For moving objects, the quantity of work/time enters calculations as distance/time, or velocity. Thus, at any instant, the rate of the work done by a force (measured in joules/second, or watts) is the scalar product of the force (a vector) with the velocity vector of the point of application. This scalar product of force and velocity is called instantaneous power. Just as velocities may be integrated over time to obtain a total distance, by the fundamental theorem of calculus, the total work along a path is similarly the time-integral of instantaneous power applied along the trajectory of the point of application.

Work is the result of a force on a point that moves through a distance. As the point moves it follows a curve X with a velocity v at each instant. The small amount of work δW that occurs over an instant of time δt is given by

where the F.v is the power over the instant δt. The sum of these small amounts of work over the trajectory of the point yields the work,

This integral is computed along the trajectory of the particle and is therefore said to be path dependent.

If the force is always directed along this line and the magnitude of the force is F, then this integral simplifies to

where s is distance along the line. If F is constant in addition to being directed along the line, then the integral simplifies further to

where d is the distance travelled by the point along the line.

This calculation can be generalized for a constant force that is not directed along the line followed by the particle. In this case the dot product F·dx = Fcosθdx, where θ is the angle between the force vector and the direction of movement, that is

In the notable case of a force applied to a body always in at an angle of 90 degrees from the velocity vector (as when a body moves in a circle under a central force) no work is done at all, since the cosine of 90 degrees is zero. Thus, no work can be done by gravity on a planet in a circular orbit (this is an ideal, as all orbits are slightly ellipical), and no work is done on a body moving in a circle at constant speed while constrained by mechanical force, such as moving at constant speed in a frictionless ideal centrifuge.

Calculating the work as "force times straight path segment" can only be done in the simple circumstances described above. If the force is changing, if the body is moving along a curved path, possibly rotating and not necessarily rigid, then only the path of the application point of the force is relevant for the work done, and only the component of the force parallel to the application point velocity is doing work (positive work when in the same direction, and negative when in the opposite direction of the velocity). This component of the force can be described by the scalar quantity called scalar tangential component (, where is the angle between the force and the velocity). And then the most general definition of work can be formulated as follows:

Work of a force is the line integral of its scalar tangential component along the path of its application point.