Determining Potential Energy Values — Question 10

Question 10

Reconstructing a two-coordinate potential

A planar conservative force has components $F_x=-2x-3y$ and $F_y=-3x-4y$ in newtons when x,y are in meters. Set U(0,0)=0.

1. Integrate the x component to find the most general intermediate form of U.

2. Use the y component to finish determining U.

3. Find U(1,2) and work from (0,0) to (1,2).

Question 10 - Solution

Reconstructing a two-coordinate potential

(a) Treat y as constant only during the x integration.

Integrating the x component with y held fixed gives $$\frac{\partial U}{\partial x}=2x+3y\Rightarrow\boxed{U=x^2+3xy+f(y)}.$$ The integration remainder may depend on y.

(b) Match the other force component.

Match the y component: $$\frac{\partial U}{\partial y}=3x+f'(y)=3x+4y\Rightarrow f(y)=2y^2+C.$$ The origin reference fixes $C=0$, hence $$\boxed{U=x^2+3xy+2y^2}.$$

(c) Evaluate the completed potential.

At the requested point, $$\boxed{U(1,2)=1+6+8=15\ \mathrm J},\qquad\boxed{W_{(0,0)\to(1,2)}=0-15=-15\ \mathrm J}.$$ Omitting $f(y)$ would recover $F_x$ but give the wrong $F_y$.