Determining Potential Energy Values — Question 6

Question 6

Recovering a quadratic potential from state measurements

A potential is known to be $U(x)=ax^2+bx+c$. Measurements give U(0)=3 J, U(1 m)=4 J and U(2 m)=9 J.

1. Find a, b and c with units.

2. Find the associated force at x=1 m.

3. Predict work from x=0 to x=3 m and state which modeling assumption permits the prediction.

Question 6 - Solution

Recovering a quadratic potential from state measurements

(a) Solve the three state equations.

Substitute the three measurements, using SI numerical coordinates: $$c=3,\qquad a+b=1,\qquad4a+2b=6.$$ Subtract twice the second relation from the third: $$\boxed{a=2\ \mathrm{J/m^2},\quad b=-1\ \mathrm{J/m},\quad c=3\ \mathrm J}.$$

(b) Differentiate the inferred function.

The force is the negative derivative: $$F_x=-\frac{d}{dx}(2x^2-x+3)=1-4x\Rightarrow\boxed{F_x(1)=-3\ \mathrm N}.$$

(c) Extrapolate only under the stipulated form.

Extrapolate the stipulated quadratic: $$U(3)=2(9)-3+3=18\ \mathrm J,\qquad\boxed{W_{0\to3}=3-18=-15\ \mathrm J}.$$ Three measurements alone would not justify this extrapolation without the assumed functional form.