SHM Period and Frequency: Real-World Examples

📚 Quick Study Guide

• ⏱️ Period (T): The time taken for one complete oscillation. Measured in seconds (s).
• 🧮 Frequency (f): The number of oscillations per unit time. Measured in Hertz (Hz), where 1 Hz = 1 oscillation per second.
• 🔗 Relationship: Period and frequency are inversely related: $f = \frac{1}{T}$ and $T = \frac{1}{f}$.
• ⚖️ SHM Condition: Restoring force is directly proportional to the displacement from equilibrium: $F = -kx$, where $k$ is the spring constant.
• 🌱 Examples: A pendulum swinging with a small angle, a mass attached to a spring oscillating on a frictionless surface, and the vibration of atoms in a solid.
• 💡 Formulas for Mass-Spring System:
  • Period: $T = 2\pi \sqrt{\frac{m}{k}}$, where $m$ is mass and $k$ is the spring constant.
  • Frequency: $f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$
• 🧭 Formulas for Simple Pendulum:
  • Period: $T = 2\pi \sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity.
  • Frequency: $f = \frac{1}{2\pi} \sqrt{\frac{g}{L}}$

🧪 Practice Quiz

1. A mass-spring system oscillates with a period of 2 seconds. What is its frequency?
   1. 0.25 Hz
   2. 0.5 Hz
   3. 1 Hz
   4. 2 Hz
2. A simple pendulum has a length of 1 meter and oscillates on Earth (g ≈ 9.8 m/s²). What is its approximate period?
   1. 0.32 s
   2. 2.0 s
   3. 4.0 s
   4. 6.3 s
3. If the frequency of a spring-mass system is doubled, what happens to the period?
   1. It doubles.
   2. It halves.
   3. It remains the same.
   4. It quadruples.
4. Which of the following is NOT an example of Simple Harmonic Motion (SHM)?
   1. A swinging pendulum with a small angle.
   2. A bouncing ball on the floor.
   3. A mass attached to a spring oscillating on a frictionless surface.
   4. The vibration of atoms in a solid.
5. If the mass attached to a spring is quadrupled, what happens to the period of oscillation?
   1. It doubles.
   2. It halves.
   3. It quadruples.
   4. It is quartered.
6. A grandfather clock relies on a pendulum to keep time. If the clock is running slow, how should you adjust the length of the pendulum to correct the time?
   1. Lengthen the pendulum.
   2. Shorten the pendulum.
   3. Increase the mass of the pendulum bob.
   4. Decrease the mass of the pendulum bob.
7. A spring with a spring constant $k$ is attached to a mass $m$. If both $k$ and $m$ are doubled, what happens to the frequency of oscillation?
   1. It doubles.
   2. It halves.
   3. It remains the same.
   4. It quadruples.