Question

A potential energy function for system 1 is given by u1(x) = cx2 + bx3. the potential energy function for system 2 is given by u2(x) = a + cx2 + bx3, where a is a positive quantity. how does the force on system 1 relate to the force on system 2 at a given position?

Answer 1

By definition,
$F = - \frac{dU}{dx}$
where U =  potential energy
F =  the force derived from the potential energy (PE).

For system 1, the PE is
U₁(x) = cx² + bx³
Therefore the force is
F₁(x) = -(2cx + 3bx²)

For system 2, the PE is
U₂(x) = a + cx² + bx³
Therefore the force is
F₂(x) = -(2cx + 3bx²)

Answer: The two forces are the same.