AP Physics C Mechanics Questions on Energy in SHM Oscillations

📚 Quick Study Guide

🔍 Simple Harmonic Motion (SHM) occurs when the restoring force is proportional to the displacement from equilibrium: $F = -kx$. 💡 The total mechanical energy in SHM is conserved if there are no non-conservative forces (e.g., friction). This means the sum of potential and kinetic energy is constant. 📝 Potential energy in SHM is given by $U = \frac{1}{2}kx^2$, where $x$ is the displacement from equilibrium and $k$ is the spring constant. ⚛️ Kinetic energy in SHM is given by $K = \frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the velocity. 📈 Total Energy $E = U + K = \frac{1}{2}kA^2$, where A is the amplitude of the oscillation. This value is constant and represents the maximum potential energy (at maximum displacement) or the maximum kinetic energy (at equilibrium). ⏱️ The angular frequency $\omega = \sqrt{\frac{k}{m}}$ relates to the period $T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}$. 📊 Velocity at any point in SHM: $v = \pm \omega \sqrt{A^2 - x^2}$. Maximum velocity occurs at equilibrium ($x=0$) and is given by $v_{max} = A\omega$.

🧪 Practice Quiz

1. A mass-spring system oscillates with simple harmonic motion. If the amplitude of the oscillation 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 particle is undergoing SHM with amplitude A. At what displacement from the equilibrium position is the potential energy equal to the kinetic energy?
   1. A) 0
   2. B) A/4
   3. C) A/2
   4. D) $A/\sqrt{2}$
3. A block of mass m is attached to a spring with spring constant k. If the mass is displaced from equilibrium and released, what is the maximum kinetic energy of the block during its oscillation?
   1. A) $\frac{1}{4}kA^2$
   2. B) $\frac{1}{2}kA^2$
   3. C) $kA^2$
   4. D) $2kA^2$
4. A simple harmonic oscillator has a period T. If the mass of the oscillator is doubled, what happens to the period?
   1. A) It remains the same.
   2. B) It is doubled.
   3. C) It is multiplied by $\sqrt{2}$.
   4. D) It is halved.
5. The potential energy of a simple harmonic oscillator is at its maximum when:
   1. A) The velocity is at its maximum.
   2. B) The displacement is zero.
   3. C) The acceleration is zero.
   4. D) The displacement is at its maximum.
6. A mass oscillates on a spring with a frequency *f*. If the spring constant is quadrupled, what is the new frequency?
   1. A) f/4
   2. B) f/2
   3. C) 2f
   4. D) 4f
7. A mass-spring system has a total energy E. If both the amplitude and the spring constant are doubled, what is the new total energy?
   1. A) E
   2. B) 2E
   3. C) 4E
   4. D) 8E