CM-Work Done By A Constant Force: Definition In One, Two, Or Three Dimensions

As stated on Page 1, the definition of CM-work W_CM done by a force is analogous to that of work. We just need to substitute the displacement _CM of the CM for the displacement of the point at which the force is acting. Thus, for a constant force,

W_CM = _CM = Fd_CM cos q

where d_CM is the magnitude of _CM and q is the angle between the force and displacement vectors.

The scalar product can be evaluated in terms of the scalar components of the vectors. In two dimensions this becomes

W_CM = _CM = F_x d_CM,x + F_y d_CM,y,

In three dimensions, the term F_z d_CM,z needs to be added on the right-hand side. For motions in one dimension along an x-axis, only the term F_x d_CM,x is needed on the right-hand side.

Example. Figure 1 below shows a five-sided object at the initial and final moments of a time interval for which we want to calculate the CM-work done on the object. The figure shows a constant force , the path of the point on which the force is acting, the displacement of this point, and the displacement _CM of the CM. The latter is shown in green.

Figure 1

As the figure illustrates, the displacement of the CM is different from the displacement of the point at which the force is acting. Therefore, the CM-work done by the force is different from the work in this case.

Comment.

cannot be 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 CM-work done by just is calculated. We could calculate the CM-work done by the other forces in the same fashion.

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

CM-Work Done By A Variable Force

If a force acting on an object or system is varying during some time interval, the CM-work done by the force during this time interval on the object or system is the sum of all the infinitesimal amounts dW_CM of CM-work done during the infinitesimal intervals that make up the finite interval. (A sum of infinitesimals is called an integral.) The path taken by the CM will in general be curved. The infinitesimal amounts of work dW_CM need to be calculated for the infinitesimal displacements of the CM that make up this curved path.

The infinitesimal CM-work dW_CM done by a variable force on an object or system during an infinitesimal time interval while the object's or system's CM undergoes the infinitesimal displacement d is defined by

dW_CM = d.