AP Physics C questions on Energy Diagrams and Turning Points

📚 Quick Study Guide

🔍 Energy Diagrams: Represent potential energy $U(x)$ as a function of position $x$. The total mechanical energy $E$ is a horizontal line on the diagram.
• 📈 Potential Energy: $U(x)$ represents the potential energy of the system. The force is related to the potential energy by $F(x) = -\frac{dU}{dx}$.
• kinetica energy $K = E - U(x)$.
• ↩️ Turning Points: Points where the kinetic energy is zero, i.e., $E = U(x)$. At these points, the object momentarily stops and changes direction.
• ⚖️ Equilibrium Points: Points where the force is zero, i.e., $\frac{dU}{dx} = 0$. These can be stable (potential energy minimum), unstable (potential energy maximum), or neutral (constant potential energy).
• 💡 Stable Equilibrium: A small displacement from equilibrium results in a restoring force pushing the object back towards equilibrium.
• 💥 Unstable Equilibrium: A small displacement from equilibrium results in a force pushing the object further away from equilibrium.

Practice Quiz

1. A particle moves along the x-axis under the influence of a potential energy function $U(x) = ax^2$, where $a$ is a positive constant. What is the force acting on the particle at position $x$?
   1. $F(x) = -2ax$
   2. $F(x) = 2ax$
   3. $F(x) = ax^2$
   4. $F(x) = -ax^2$
2. The potential energy of a particle is given by $U(x) = x^3 - 6x^2 + 5$. At what position(s) is the particle in equilibrium?
   1. $x = 0$ and $x = 4$
   2. $x = 0$ and $x = -4$
   3. $x = 0$
   4. $x = 4$
3. A particle with total energy $E$ is moving in a potential $U(x)$ as shown in an energy diagram. If $E < U_{max}$, the particle's motion is:
   1. Unbounded
   2. Periodic between two turning points
   3. Uniform
   4. Constantly accelerating
4. Given a potential energy function $U(x) = -\frac{a}{x} + bx^2$, where $a$ and $b$ are positive constants, find the equilibrium position.
   1. $x = \sqrt[3]{\frac{a}{2b}}$
   2. $x = \sqrt{\frac{a}{b}}$
   3. $x = \frac{a}{b}$
   4. $x = \sqrt[3]{\frac{2b}{a}}$
5. A particle's potential energy is described by $U(x) = 4x^4$. What is the force acting on the particle?
   1. $F(x) = -16x^3$
   2. $F(x) = 16x^3$
   3. $F(x) = 4x^4$
   4. $F(x) = -4x^4$
6. A particle of mass $m$ moves in a potential energy $U(x)$. At a turning point, what is its kinetic energy $K$?
   1. $K = 0$
   2. $K = U(x)$
   3. $K > 0$
   4. $K < 0$
7. For the potential energy function $U(x) = x^4 - 2x^2$, identify the stable equilibrium point(s).
   1. $x = -1$ and $x = 1$
   2. $x = 0$
   3. $x = -1, x = 0, x = 1$
   4. There are no stable equilibrium points.