AP Physics Questions on Energy in Simple Harmonic Motion

📚 Quick Study Guide

• 📏 Simple Harmonic Motion (SHM): A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.
• 💡 Total Energy in SHM: The total mechanical energy ($E$) of a system in SHM is the sum of its kinetic energy ($K$) and potential energy ($U$). $E = K + U$
• 🔄 Potential Energy: For a spring-mass system, the potential energy is given by $U = \frac{1}{2}kx^2$, where $k$ is the spring constant and $x$ is the displacement from equilibrium.
• 🏃 Kinetic Energy: The kinetic energy is given by $K = \frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the velocity.
• ⭐ Total Energy (Alternative): $E = \frac{1}{2}kA^2$, where $A$ is the amplitude of the motion.
• ⏱️ Angular Frequency: $\omega = \sqrt{\frac{k}{m}}$, relates to the period $T$ by $T = \frac{2\pi}{\omega}$.

🧪 Practice Quiz

1. A mass-spring system oscillates with simple harmonic motion. If the amplitude is doubled, what happens to the total energy of the system?
   1. A) It remains the same.
   2. B) It is doubled.
   3. C) It is quadrupled.
   4. D) It is halved.
2. A block of mass $m$ is attached to a spring with spring constant $k$. The system oscillates with amplitude $A$. What is the maximum kinetic energy of the block?
   1. A) $\frac{1}{4}kA^2$
   2. B) $\frac{1}{2}kA^2$
   3. C) $kA^2$
   4. D) $2kA^2$
3. At what displacement from the equilibrium position is the potential energy equal to the kinetic energy in simple harmonic motion?
   1. A) $x = A$
   2. B) $x = \frac{A}{2}$
   3. C) $x = \frac{A}{\sqrt{2}}$
   4. D) $x = 0$
4. A simple pendulum oscillates with a small amplitude. If the length of the pendulum is quadrupled, what happens to the period?
   1. A) It remains the same.
   2. B) It is halved.
   3. C) It is doubled.
   4. D) It is quadrupled.
5. A mass-spring system oscillates vertically. How does the gravitational potential energy affect the total energy calculation in SHM?
   1. A) It increases the total energy.
   2. B) It decreases the total energy.
   3. C) It shifts the equilibrium position but does not affect the total energy.
   4. D) It dampens the oscillations.
6. If the total energy of a simple harmonic oscillator is increased by a factor of 9, by what factor does the amplitude increase?
   1. A) 3
   2. B) 4.5
   3. C) 9
   4. D) 81
7. A spring with spring constant $k$ is compressed a distance $x$ from its equilibrium position. What is the work required to compress the spring?
   1. A) $\frac{1}{4}kx^2$
   2. B) $\frac{1}{2}kx^2$
   3. C) $kx^2$
   4. D) $2kx^2$