Calculating work done by a force

LEARNING OBJECTIVE 3.2.A - Describe the work done on an object or system by a given force or collection of forces.
AP MECHANICS -C LEARNING OBJECTIVE INT 4.A -Calculate work done by a given force (constant or as a given function F(x)) on an object that undergoes a specified displacement.

let’s now define the work mathematically, so that we can calculate its value.

To introduce the definition of work done by a force, we begin with the simple case of motion along a straight line, with the force along the line of motion. Consider a constant force F acting on a particle due to which the particle moves a distance $\Delta$x along a straight line. Then the work done by the force $F_x$ on the particle as it moves some given distance $\Delta x$ is defined as the product of the force and the displacement. We are using $\Delta$ to signify the change in position due to the action of the force.

$W = F_x \Delta x$

Though this equation is pretty simple, it has several silent points, we must consider.

• when the point of application of a force moves, the force does work. If, however, the point of application of a force does not move in spite of the action of the force, no work is done. For example, If a load hanging on a suspender is at rest, the force of gravity acting on the load does no work. If, however, the load descends or falls, this force does a work equal to mgh (mg is the force of gravity and h is the distance by which the load has been lowered. Look exactly where the force is applied and if that particular point moves or not.
• Work is a number ( rather than a vector). Since the magnitude of the force F is also a number. It must be either a positive number or zero. However, the displacement along the force can be either positive, negative or zero, depending upon the direction of displacement relative to the force. Hence, work could also be positive, negative or zero.
• The work is positive if the force and the displacement are in the same direction (both positive, or both negative), and the work is negative if they are in opposite directions (one positive, the other negative)
• For example, if you are pushing your car in the direction of its movement, the work done by you is positive. But if you are pushing the car in the opposite direction of its displacement the work done by you is negative.
• A complete description of work requires specifying three things. First, the particle on which the work is getting done. Secondly, what force is doing the work and finally the path of displacement.
• That is to say, ” Work done on the particle by a force along the path “. All three are required to describe the work completely.

Work done on the particle by a force along the path. All three are required to describe the work completely.

Defining work