Solved Examples of Finding Equilibrium from Potential Energy Curves

📚 Quick Study Guide

• 📉 Equilibrium points are where the slope of the potential energy curve is zero: $\frac{dU}{dx} = 0$.
• stable equilibrium corresponds to a minimum in the potential energy curve. Think of it like a ball at the bottom of a bowl.
• ⛰️ Unstable equilibrium corresponds to a maximum in the potential energy curve. Imagine a ball balanced at the very top of a hill.
• ↔️ Neutral equilibrium occurs where the potential energy is constant over a region. A ball on a flat surface represents this.
• ➗ Force is related to the potential energy by $F = -\frac{dU}{dx}$. The negative sign indicates that the force points in the direction of decreasing potential energy.
• 💡To find equilibrium points, take the derivative of the potential energy function with respect to position (x) and set it equal to zero.
• 📝 Analyze the second derivative, $\frac{d^2U}{dx^2}$, to determine the stability of the equilibrium. If it's positive, it's stable; if it's negative, it's unstable; if it's zero, further analysis is needed.

Practice Quiz

1. A particle's potential energy is given by $U(x) = 3x^2 - x^3$. At what position(s) is the particle in equilibrium?

   1. A) $x = 0$ only
   2. B) $x = 0$ and $x = 2$
   3. C) $x = 2$ only
   4. D) $x = 3$ and $x = -3$

2. For the potential energy function $U(x) = x^4 - 2x^2$, identify the positions of stable equilibrium.

   1. A) $x = 0$
   2. B) $x = 1$ and $x = -1$
   3. C) $x = 0$, $x = 1$, and $x = -1$
   4. D) $x = \sqrt{2}$ and $x = -\sqrt{2}$

3. Given the potential energy curve where $U(x)$ is constant between $x = a$ and $x = b$, what type of equilibrium exists in this region?

   1. A) Stable equilibrium
   2. B) Unstable equilibrium
   3. C) Neutral equilibrium
   4. D) Dynamic equilibrium

4. If the potential energy function is $U(x) = -\frac{k}{x}$ (where k is a positive constant), what can you say about the equilibrium?

   1. A) Stable equilibrium at $x = 0$
   2. B) Unstable equilibrium at $x = 0$
   3. C) There is no equilibrium point
   4. D) Neutral equilibrium for all x

5. The potential energy of a system is described by $U(x) = e^{-x^2}$. Where is the equilibrium point located?

   1. A) $x = 1$
   2. B) $x = -1$
   3. C) $x = 0$
   4. D) There are no equilibrium points.

6. Consider a potential energy curve. At a point where the curve has a local maximum, what type of equilibrium is present?

   1. A) Stable
   2. B) Unstable
   3. C) Neutral
   4. D) Dynamic

7. If a particle is displaced slightly from a position of stable equilibrium, what will it tend to do?

   1. A) Move further away from the equilibrium point
   2. B) Return to the equilibrium point
   3. C) Move to a new equilibrium point
   4. D) Oscillate with increasing amplitude