Unit 7: Oscillations

Simple Harmonic Motion (SHM)

• Oscillatory motion where a **restoring force** acts towards equilibrium.
• Restoring force is **proportional** to displacement from equilibrium.
• Mathematically: $$F = -k x$$

Mass-Spring System

• The system moves through positions in a repeating pattern (e.g., **1, 2, 3, 2, 1, 2, 3, 2, 1**).
• Key positions:
  • Position 1: $x = A$, $v = 0$, $|a| = \text{maximum}$.
  • Position 2: $x = 0$, $|v| = \text{maximum}$, $a = 0$.
  • Position 3: $x = -A$, $v = 0$, $|a| = \text{maximum}$.

Derivation of SHM Equations

By comparing SHM to **circular motion**, we derive the equation:

$$\cos\theta = \frac{x}{r} \Rightarrow x = r \cos\theta$$

Using the relationship: $$T = \frac{2\pi}{\omega}, \quad \omega = 2\pi f$$

And knowing that: $$\theta = \omega t$$

We get the equation of motion for SHM: $$x = A \cos(2\pi f t)$$

Period of Oscillation

• For a **mass-spring system**: $$T_s = 2\pi \sqrt{\frac{m}{k}}$$
  • **Independent** of amplitude and acceleration due to gravity.
• For a **pendulum**: $$T_p = 2\pi \sqrt{\frac{L}{g}}$$
  • **Independent** of amplitude and mass.