Contents

Page 1. General Remarks And Definition Of Work In One Dimension For A Constant Force

Page 2. Definition of Work In One, Two, Or Three Dimensions For A Constant Force

Page 3. Definition of Work For A Variable Force

Work: Generalities

Work is a transfer of mechanical energy to a system. Like energy, work is a scalar quantity. The SI-unit of work, like that of energy, is the joule (J).

When mechanical energy is being transferred to a system, one says that work is being done on the system. For work to be done on a system, there must be a force acting on the system, and one says that the force is doing work on the system. The work done by a force can be either positive, negative or zero. Positive work being done on a system means that energy is being added to the system, and negative work being done on the system means that energy is being taken away from the system.

Work done by a constant force: simplest case (one dimension)

Figure 1

Figure 1 illustrates a situation in which a constant force is acting on a green block during a time interval during which the point (round black dot) at which the force is acting on the block undergoes a displacement in the direction of the force. The work W done on the block by this force is defined as

W = Fd, (1)

i.e., as the product of the magnitude F of the force and the magnitude d of the displacement.

Figure 2 below illustrates a similar situtation, but this time the block's displacement is opposite to the direction of the force. (Several circumstances can make this possible. E.g., the initial velocity of the block could be to the left and the applied force might be slowing the block down while the block continues moving left. Or there could be other forces besides acting on the block that are not shown in Figure 2 and are directed to the left.)

Figure 2

In this case, the work done by force on the block is defined as

W = -Fd, (2)

i.e., as the negative of the expression in Equation (1).

Comment. The point at which the force is acting plays a crucial role in this definition. It is the displacement of this point that matters in the definition of work, not the displacement of the object. In the case of a rigid block that is not rotating, the displacement of the point at which the force is acting is also the displacement of the block itself and of every part of the block. However, when a rigid object is rotating or when an object is deformable and changing shape, different parts of the object will undergo different displacements.