Examples of SHM: Simple Pendulum vs. Spring System

📚 Quick Study Guide

⏱️ Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. 📏 Displacement (x): Distance from the equilibrium position. 🚀 Amplitude (A): Maximum displacement from equilibrium. 🔄 Period (T): Time for one complete oscillation. frequency (f): Number of oscillations per unit time ($f = \frac{1}{T}$).

⚖️ A mass (bob) suspended from a fixed point by a string or rod. 📐 Restoring Force: Component of gravity pulling the bob towards the equilibrium. 🧮 Period Formula: $T = 2\pi \sqrt{\frac{L}{g}}$, where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity. 💡 Key Points: The period is independent of the mass of the bob and depends only on the length and gravity.

⚙️ A mass attached to a spring that can oscillate back and forth. 💪 Restoring Force: Provided by the spring, obeying Hooke's Law. 📜 Hooke's Law: $F = -kx$, where $k$ is the spring constant. 🧮 Period Formula: $T = 2\pi \sqrt{\frac{m}{k}}$, where $m$ is the mass and $k$ is the spring constant. 💡 Key Points:** The period depends on the mass and the spring constant. Stiffer springs (higher $k$) result in shorter periods.

Practice Quiz

1. A simple pendulum has a length of 1 meter. What is its approximate period (assuming g = 9.8 m/s²)?

   • A) 1 second
   • B) 2 seconds
   • C) 3 seconds
   • D) 4 seconds

2. If the length of a simple pendulum is quadrupled, what happens to its period?

   • A) It doubles
   • B) It halves
   • C) It remains the same
   • D) It quadruples

3. A mass of 2 kg is attached to a spring with a spring constant of 8 N/m. What is the period of oscillation?

   • A) $\pi$ seconds
   • B) $2\pi$ seconds
   • C) $\frac{\pi}{2}$ seconds
   • D) $\frac{\pi}{4}$ seconds

4. If the mass attached to a spring is doubled, what happens to the period of oscillation?

   • A) It doubles
   • B) It halves
   • C) It increases by a factor of $\sqrt{2}$
   • D) It decreases by a factor of $\sqrt{2}$

5. Which of the following factors does NOT affect the period of a simple pendulum?

   • A) Length of the pendulum
   • B) Acceleration due to gravity
   • C) Mass of the bob
   • D) Angle of displacement (for small angles)

6. What happens to the period of a spring-mass system if the spring constant is increased?

   • A) It increases
   • B) It decreases
   • C) It remains the same
   • D) It doubles

7. In SHM, at what point is the velocity of the oscillating object the greatest?

   • A) At the maximum displacement
   • B) At the equilibrium position
   • C) At half the maximum displacement
   • D) Velocity is constant throughout the motion