# The potential energy function for a system of particles is given by U(x) = -3x^3 + 4x^2 + 3x,...

## Question:

The potential energy function for a system of particles is given by {eq}U(x) = -3x^3 + 4x^2 + 3x {/eq}, where x is the position of one particle in the system.

(a) Determine the force Fx on the particle as a function of x.

(b) For what values of x is the force equal to zero? (Enter your answers from smallest to largest.)

## Potential Energy in Terms of Force:

Potential energy of an object is the energy possessed by the object by virtue of its configuration or position.

If the force on an object is the function of position only then it can be represented in terms of potential energy as

{eq}\vec{F}=-\dfrac{\mathrm{d} \ U}{\mathrm{d} \vec{r}}\\ {/eq}

Or if the force is along one direction only, say x direction, then

{eq}F(x)=-\dfrac{\mathrm{d} \ U(x)}{\mathrm{d} x} {/eq}

- {eq}\vec{F} {/eq} = force.

- {eq}U {/eq} = potential energy.

## Answer and Explanation:

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**Given:**

- {eq}U(x) = -3x^3 + 4x^2 + 3x {/eq}

**Part (a):**

{eq}F(x)=-\dfrac{\mathrm{d} \ U(x)}{\mathrm{d} x}\\ =-\dfrac{\mathrm{d} (-3x^3 + 4x^2 +...

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Chapter 5 / Lesson 8Learn how to tell if a force is conservative and what exactly is being conserved. Then look at a couple of specific examples of forces to see how they are conservative.