Simple Harmonic Motion (SHM)

Vertical Spring

• Unstretched Length: Initial state of the spring without mass.
• Stretched Length: When mass is added, spring stretches downward.
  • Displacement: Measured as X1 for initial mass.
  • Increased Displacement: X2 for doubled mass.
• Force vs. Displacement Graph:
  • Linear graph with slope K, the spring constant.
  • Spring Constant (K):
    • Indicator of stiffness:
      • Large K: Stiff spring.
      • Small K: Stretchy spring.
  • Hook's Law: F = -K * X
    • Restoring Force: Opposite direction of displacement.
    • Undamaged Spring Assumption: Complies with Hook's Law.
• Energy in Spring:
  • Area Under Graph: Represents stored energy.
  • Energy Equation: 1/2 * K * X^2
    • Derived using triangle area formula and work equation.*

Work and Energy Relationship

• Work Equation:
  • Average Force Method: 1/2 * K * X * X = 1/2 * K * X^2
  • Work = Change in Energy: Work done equals increase in potential energy.*

Simple Harmonic Motion (SHM) Concepts

• Pendulum & Mass on Spring:
  • Period (T): Time for one complete cycle.
    • Calculation: Total time divided by number of cycles.
  • Frequency (f): Number of cycles per second.
    • Calculation: Cycles divided by total time.
  • Relationship: T = 1/f and f = 1/T

Harmonic Oscillator Analysis

• Mass on Spring: Experiences restoring force proportional to displacement.
  • Force Diagram:
    • At Equilibrium: No restoring force (net force is zero).
    • At Amplitude: Zero velocity, maximum acceleration.
  • Acceleration (a): Maximum at amplitude due to maximum restoring force.
  • Amplitude Impact on Period: Period is independent of amplitude.
  • Period Equation for Spring: T = 2π * √(m/K)
  • Frequency Equation: f = 1/(2π) * √(K/m)

Simple Pendulum

• Restoring Force: Due to weight component.
  • For Small Angles (θ < 15°): sin(θ) ≈ θ
• Pendulum Period:
  • Equation: T = 2π * √(L/g)
    • L: Length of pendulum.
    • g: Acceleration due to gravity (9.8 m/s²).*

Energy in Horizontal Spring

• Total Mechanical Energy: Sum of potential and kinetic energy, remains constant.
  • Elastic Potential Energy: At maximum when spring is stretched or compressed.
  • Kinetic Energy: Maximum at equilibrium.
• Energy Transfer:
  • Kinetic to Potential and vice versa: During motion.
  • Energy Equations:
    • Potential Energy: 1/2 * K * A^2
    • Kinetic Energy: 1/2 * m * v_max^2
  • Solving for Unknowns: e.g., finding maximum velocity with given parameters.