Work and energy in three dimensions

As we already know, work is defined as the scalar product of the force and displacement vector. Thus if a particle moves under the influence of a force field from point 1 to point 2 along the path shown below, the total work is calculated as the sum of partial work done when the particle moves a vanishingly small distance along the arrows shown below in the figure.

Thus the total work done in gives as

where C(12) indicates that the particle is moving along the curve C from point 1 to 2 . Writing the dot product explicitly, we get

where F_i (i = x,y,z) indicates the i^th component of the force and x, y and z are varied along the curve. Let us do an example of calculating the work in this manner in two-dimensions.

Example: Consider two force fields (a) , and (b) in the x-y plane. Calculate the work done by these forces when a particle moves from the origin to (1,2) along the three paths C1, C2 and C3 shown in the figure below. On C1 the particle goes along the x-axis first and then moves parallel to the y-axis; on C2 it travels along the y-axis first and then parallel to the x-axis and on C3 it moves along the diagonal.

The work done is given by the formula

Along C1 y = 0 , dy = 0 while moving along the x-axis whereas x = 1 and dx = 0 when the particle travels parallel to the y-axis. Thus the work done along C1 is

Similarly work done along C2 is given as

For path C3, we have y = 2x so that dy = 2dx . Therefore we substitute y = 2x in the functions giving the force and replace dy by 2dx . As a result, the final integration is over x only with x varying from 0 to 1 . Thus the work done is

We are now ready to work out the work done by force in (a) and (b) (I would like you to plot these force fields and leave it as an exercise for you). For the force in (a) we get

;

;

For force (b) on the other hand we get

Thus we see that whereas the force in (a) gives the work to be the same for all three paths, that in (b) gives different work along the three paths. Thus the first force field may be conservative but the second one is definitely not.

Now let us derive the work-energy theorem in three dimensions. Start from the equation of motion and take the dot product of both sides with the velocity to get

Now integrate both sides with respect to time and use , where is the small distance traveled by the particle in time interval dt , to get

On integration this leads to

This equation tells us that when a force makes a particle move along path C from point 1 to 2, the work done by the force equals the change in its kinetic energy. This is the work-energy theorem in three-dimensions. It is exactly the same as in one dimension except that the work done is calculated by moving along a three-dimensional path.

Potential energy: As is the case in one dimensional motion, potential energy in general can be defined only if the work done is path independent. In that case, the work done depends only on the end points of the path of travel and can be written as the difference on a quantity that is a function of the position vector only. Thus

where is defined as the potential energy. Notice that this time I have not written any specific path but just the end points with the integral sign because the work is supposed to be path-independent. From the definition above, it is also evident that here too the difference in the potential energy between point 1 and point 2 is the work done by us in moving a particle slowly, maintaining its equilibrium, from point 1 to point 2. Now following the exactly same steps that we did for the one dimensional case, we show that

Thus when the potential energy can be defined, the total mechanical energy of a particle is conserved . I remind you that the total mechanical energy is the sum of the kinetic and the potential energies. In such cases the force is said to be conservative.

By now you may be wondering how can we find out whether a force is conservative or not. Do we have to calculate the work done along all possible paths before we can say that the force is conservative and therefore the principle of conservation of energy holds good. That certainly would be impossible to do. However, there is a much simpler test to check whether a force field is conservative or not. I am going to tell you about it without giving the proof. To find out about the conservative nature of a force , we calculate its curl defined as

Now if the curl of a force field vanishes everywhere, it is a conservative force field. On the other hand if the curl of a force field is nonzero, it is not conservative. Let us now apply this test to the two force fields for which we calculated the work done along different paths. For the force field , the curl is zero everywhere. Hence it is conservative and, as we saw with three paths, the work done in this field is indeed path independent. On the other hand, for , the curl comes out to be and therefore the force is not conservative. This was seen above where the work done along the three paths were all different. We now solve an example where knowing the conservative nature beforehand helps us avoid an unnecessary calculation.