Saturday, May 3, 2008

Irodov Problem 1.133

Force is the negative gradient of potential i.e. in catersian coordinates we have,

.

In this problem the question is whether or not there exists some potential function U whose negative gradient is the force F.

a) From the problem we have,







From (1b) and (1c) it is clear that U = f(x) i.e. U is only a function of x and not of y and z. If however, this were true then, (1a) is impossible since it implies that U is of the form U = axy + f1(y,z). This contradiction implies that F cannot be a potential function.

b)From the problem we have,







From (2c) clearly U = f(x,y). From (2a), we have that the potential U has the form . From (2c) the potential can be written as . Based on these observation we can see that a potential function that satisfies all above conditions is,
.


In other words there is a potential function that can result in the above force.

2 comments:

Rudraneil said...

wh at is exactly the significance of the curl?

Krishna Kant Chintalapudi said...

curl measures rotation of a vector field at any point. The easiest way to understand curl is via fluid flows. For example, consider a flowing fluid. Let us consider the velocities of different points of the fluid as a vector field. Then if the curl of this vector field is zero, it means that the fluid is not spinning. In fact curl of velocity is called vorticity. In vortices curl will be non-zero.