AP Physics C: Mechanics Questions on Potential Energy Curves and Equilibrium

📚 Quick Study Guide

⚛️ Potential Energy (U) is the energy an object has due to its position. It's related to the conservative forces acting on the object. 📊 Potential Energy Curves plot potential energy (U) as a function of position (x). The slope of the curve relates to the force. 📉 Force (F) is the negative derivative of potential energy with respect to position: $F = -\frac{dU}{dx}$. ⚖️ Equilibrium occurs where the net force on an object is zero. On a potential energy curve, this happens at points where the slope is zero (minima, maxima, or inflection points). 安定 Stable Equilibrium occurs at a minimum on the potential energy curve. If displaced slightly, the object will return to equilibrium. 不安定 Unstable Equilibrium occurs at a maximum on the potential energy curve. If displaced slightly, the object will move away from equilibrium. neutral Neutral Equilibrium occurs where the potential energy is constant over a region. A small displacement will not change the potential energy or force. 💡 Total Mechanical Energy (E) is the sum of kinetic energy (K) and potential energy (U): $E = K + U$. ✏️ Turning Points are the locations where the kinetic energy is zero (total energy equals potential energy).

Practice Quiz

1. A particle's potential energy is given by $U(x) = 3x^2 - x^3$. At what position is the particle in equilibrium?
   1. $x = 0$ only
   2. $x = 2$ only
   3. $x = 0$ and $x = 2$
   4. $x = -2$
2. Which of the following statements is true about stable equilibrium?
   1. The potential energy is at a maximum.
   2. The force is zero, and a small displacement will cause the object to move away from equilibrium.
   3. The potential energy is at a minimum.
   4. The force is non-zero.
3. A potential energy curve is flat between x = 1m and x = 3m. What can you say about the force in this region?
   1. The force is constant and positive.
   2. The force is constant and negative.
   3. The force is zero.
   4. The force is increasing linearly.
4. The potential energy of a particle is given by $U(x) = -ax^{-2} + bx$, where a and b are positive constants. What is the equilibrium position?
   1. $x = \sqrt[3]{\frac{2a}{b}}$
   2. $x = \sqrt{\frac{a}{b}}$
   3. $x = \frac{a}{b}$
   4. $x = \sqrt[3]{\frac{a}{2b}}$
5. If the total mechanical energy of a system is E, and the potential energy at a certain point is greater than E, what can you say about the particle's motion?
   1. The particle is at a turning point.
   2. The particle's kinetic energy is negative, which is impossible.
   3. The particle is in stable equilibrium.
   4. The particle's velocity is constant.
6. A particle is in unstable equilibrium at x = 0. Which of the following best describes the potential energy curve near x = 0?
   1. The potential energy has a minimum at x = 0.
   2. The potential energy has a maximum at x = 0.
   3. The potential energy is constant near x = 0.
   4. The potential energy is linearly increasing near x = 0.
7. Given the potential energy $U(x) = x^4 - 2x^2$, find the positions of stable equilibrium.
   1. $x = 0$
   2. $x = 1$ and $x = -1$
   3. $x = 0$, $x = 1$, and $x = -1$
   4. $x = \sqrt{2}$ and $x = -\sqrt{2}$