Simple Harmonic Motion

Key Points

• Periodic Motion: Simple harmonic motion is repetitive and periodic.
• Acceleration: It is directly proportional to the negative value of the displacement.
• Restoring Force: It always acts towards the mean position.

Equations and Concepts

• Acceleration Equation: $a = -w^2 x$
  • $a$: Acceleration
  • $w$: Angular frequency
  • $x$: Displacement

Examples

Spring-Mass System

• Initial Condition: When the mass is at rest, the weight equals tension.
• Force on Extension: $F = kx$
• Relationship of Tension and Weight: If tension is greater than the weight, the resultant force is upwards.

Pendulum

• Components of Tension: $T\cos\theta$, $T\sin\theta$
• Effect of Position:
  • Minimum displacement occurs in the third position.

Review of Equations

• Displacement:

  • Starting from maximum displacement: $x = x_0 \cos(\omega t)$
  • Starting from mean position: $x = x_0 \sin(\omega t)$

• Derivation of Velocity and Acceleration:

  • Derivation of the equations for velocity and acceleration is performed.

Energy

• Kinetic Energy: $KE = \frac{1}{2} m v^2$
• Potential Energy: $PE = \frac{1}{2} m \omega^2 x^2$
• Total Energy: $TE = KE + PE = \frac{1}{2} m \omega^2 x_0^2$

Resonance

• Definition: When the natural frequency equals the driving frequency, there is an increase in amplitude.

Types of Damping

• Light Damping: Stabilizes gradually after oscillation.
• Critical Damping: Oscillation stops without a complete cycle.
• Heavy Damping: Achieves stability very slowly.

These are the important points and complexities of simple harmonic motion included in the A2 syllabus.