Work done by a constant force: definition in one, two, or three dimensions

The definition of work done on a system is illustrated here in terms of a five-sided object that undergoes both translational and rotational motion. Figure 1 below shows the object at the initial and final moments of the time interval for which we want to define the work done on the object.

Figure 1

The definition of work done by a force involves the force and the displacement of the point at which the force is acting. In general, the displacement of the point at which the force is acting is not in the same direction as the force. In Figure 1, the angle between the force and this displacement is indicated by the symbol q (Greek letter; read: theta). Whether the directions of the displacement and that of the force are the same or not, the work done by the force is defined as follows.

Definition. Suppose the force acting on an object is constant during some time interval and that the point at which the force is acting on the object undergoes a displacement during this time. Then the work W done on the object by the force is defined by

W = = Fd cos θ = F_x d_x + F_y d_y,

i.e., by the scalar product (dot product) between the force and the displacement. Here, θ is the angle between the two vectors and . If the motion is three-dimensional and the displacement has a non-zero z-component d_z, a third term F_z d_z must be added to F_x d_x + F_y d_y in the equation above. If the motion is one-dimensional and entirely along the x-axis, only the term F_x d_x is needed on the right-hand side.

The scalar product can be calculated either in terms of the angle θ and the magnitudes F and d of the two vectors or in terms of the Cartesian components of the two vectors. In the case illustrated in Figure 1, the work W is positive because the angle θ is less than 90 degrees so that its cosine is positive. If θ > 90^o, the work is negative.

Comments.

1. The point at which the force is acting is indicated by a black dot in Figure 1. The arrow representing the force is drawn with its tail end placed at this point. The stippled line indicates the path along which the point moves. The shape of this path, roughly a semi-circle in Figure 1, does not enter into the expression for the work above. The only thing that matters is the net displacement of the point of action of the force.

2. Similarly, the fact that the object undergoes a rotation during the displacement has no bearing on the work done by the force . What matters is only the displacement of the point of action of the force.

3. is not the only force that is acting on the object. (The other forces are not shown.) The object could not be moving along the stippled path if were the only force. (Why not?) The fact that other forces are present has no bearing on how the work done by just is calculated. We could calculate the work done by the other forces in the same fashion.

If we add the amounts of work done by all individual forces, we get what is called the net work done on the object.