How Do Constant Energy Lines Behave in a 2D Force Field?

[ajlucia]

Constant Energy Lines??

Homework Statement

Potential energy function for two dimensional force is given by U = Cxe^(-y) where C is constant.

a. Sketch the constant energy lines.

b. Show that if a point is displaced by a short distance dx along a constant energy line, then its total displacement must be dr_vector = dx (i_hat + j_hat / x(

c. Using the result in b, show explicitly that gradient of U is perpendicular to the constant energy line.

Thanks!

Homework Equations

The Attempt at a Solution

 

[jbsteele]

a. The constant energy lines are the contours of the potential energy function U(x,y). These can be found by setting U(x,y) = constant and plotting the resulting equations. b. Let (x_0, y_0) be a point on a constant energy line. Then, the total displacement dr_vector = dx (i_hat + j_hat / x_0), where x_0 is the x-coordinate of the point.c. Since the total displacement is perpendicular to the constant energy line, it follows that the gradient of U must also be perpendicular to the constant energy line.