Conservation of Energy in SHM: Real Life Examples

📚 Quick Study Guide

• 🎢 Total Energy: The total energy ($E$) in SHM remains constant if there are no external forces like friction. $E = \frac{1}{2}kA^2$, where $k$ is the spring constant and $A$ is the amplitude.
• 🔄 Potential Energy: Potential energy ($U$) is stored when the object is displaced from its equilibrium position. $U = \frac{1}{2}kx^2$, where $x$ is the displacement.
• 🏃 Kinetic Energy: Kinetic energy ($K$) is the energy of motion. $K = \frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the velocity.
• ⚖️ Energy Conservation: At any point in SHM, $E = K + U$. The energy continuously transforms between potential and kinetic energy.
• 📍 Equilibrium Position: At the equilibrium position ($x = 0$), potential energy is zero, and kinetic energy is maximum.
• 📍 Extreme Positions: At the extreme positions ($x = ±A$), kinetic energy is zero, and potential energy is maximum.

🧪 Practice Quiz

1. Question 1: In an ideal SHM system, what happens to the total energy if the amplitude of oscillation is doubled?
   1. A) It remains the same.
   2. B) It doubles.
   3. C) It quadruples.
   4. D) It halves.
2. Question 2: A mass-spring system oscillates with SHM. At which point is the kinetic energy maximum?
   1. A) At the maximum displacement.
   2. B) At the equilibrium position.
   3. C) When the velocity is zero.
   4. D) When the acceleration is maximum.
3. Question 3: A pendulum swings with SHM (small angles). At which point is the potential energy greatest?
   1. A) At the lowest point of the swing.
   2. B) At the highest point of the swing.
   3. C) When the pendulum is moving fastest.
   4. D) When the pendulum is passing through the equilibrium position.
4. Question 4: In SHM, if the potential energy is $\frac{1}{4}$ of the total energy, what fraction of the total energy is kinetic energy?
   1. A) $\frac{1}{4}$
   2. B) $\frac{1}{2}$
   3. C) $\frac{3}{4}$
   4. D) 1
5. Question 5: A block attached to a spring oscillates horizontally. If the spring constant is $k$ and the amplitude is $A$, what is the total energy of the system?
   1. A) $\frac{1}{4}kA^2$
   2. B) $\frac{1}{2}kA^2$
   3. C) $kA^2$
   4. D) $2kA^2$
6. Question 6: A simple pendulum is an example of SHM (for small oscillations). What form of potential energy is primarily involved?
   1. A) Elastic potential energy.
   2. B) Gravitational potential energy.
   3. C) Chemical potential energy.
   4. D) Nuclear potential energy.
7. Question 7: Consider a mass oscillating on a spring. If you increase the mass attached to the spring, what happens to the total energy (assuming the amplitude remains constant)?
   1. A) It increases.
   2. B) It decreases.
   3. C) It remains the same.
   4. D) It oscillates.