1D example of the relation between force and potential energy

For a simple one dimensional example of the relationship between force and potential energy, assume that the potential energy of a particle is given by the equation

A [ −x∕c] U (x) = − -- 1 + Be x

The realtionship between force and potential energy is

F = − ∇U

(1)

For our 1D example, where the potential energy is dependent only on the position, x

dU- F = − dx

Taking the derivative yields

[ ] dU- A--[ − x∕c] A- B- −x∕c dx = x2 1 + Be − x − C e

Therefore, the force on the particle is governed by the equation

A [ ] A [ B ] F = − -2- 1 + Be −x∕c + -- − --e−x∕c x x C

(2)

If we further assume that the constant C is so much larger than x, the force will simplify. Rearraning to get

A [ Bx ] F = − --- 1 + Be − x∕c −---e− x∕c x2 C

Because x ≪ C, e^−x∕c ⇒ e⁰ = 1, we get

[ ] F = − -A- 1 + B − Bx-- x2 C

Further simplifying BxC- = 0 gives the force under this assumption

F = − A--[1 + B ] x2

(3)

"1D example of the relation between force and potential energy" is owned by bloftin.
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