Lecture Notes: Periodic Motion, Harmonic Motion, and Mass-Spring Systems

Periodic Motion and Simple Harmonic Motion (SHM)

• Periodic Motion: Motion that repeats itself; oscillates back and forth.
  • Examples include the mass-spring system and the simple pendulum.
• Mass-Spring System: A spring attached to a mass at a horizontal equilibrium position.
  • Restoring Force: A force that pushes or pulls the mass back to the equilibrium position when displaced.
  • Hooke’s Law: F_restore = -kx
    • k: Spring constant (newtons per meter)
    • x: Distance from the equilibrium position
    • The negative sign indicates that the restoring force is always directed opposite to the displacement.

Restoring Force and Spring Constant

• Spring Constant (k): Measures stiffness of the spring.
  • High k: Stiffer spring requiring more force to stretch by a certain length.
  • Example: If k = 100 N/m, it requires 100 N to stretch the spring by 1 meter.
• Hooke’s Law Explanation: Negative sign is due to restoring force direction.
  • At Equilibrium: F_restore is zero since x = 0.
• Practical Examples:
  • Stretching a Spring: Force required to stretch a spring can be calculated using F = kx.
  • Unit Conversion: Use meters for distance to match units of the spring constant.

Work and Energy in Springs

• Work to Stretch a Spring: Work W = 0.5 kx² (or potential energy).
  • Integral Form: Accounts for variable force.
  • Potential Energy (U): U = 0.5 kx²
  • Mechanical energy is conserved in the absence of friction.

Dynamics of Mass-Spring Systems

• Equilibrium and Motion:
  • At Equilibrium: Velocity maximum, restoring force, and acceleration zero.
  • Fully Stretched/Compressed: Velocity zero, restoring force, and acceleration maximum.
  • Undamped Oscillations: Continue indefinitely in absence of friction.
  • Damped Oscillations: Amplitude decreases over time due to friction.

Energy Transfer and Mechanical Energy

• Kinetic Energy (KE): KE = 0.5 mv²
• Potential Energy (PE): Maximum when stretched/compressed.
• Mechanical Energy (ME): ME = KE + PE, remains constant in frictionless systems.

Acceleration and Velocity Calculations

• Maximum Velocity (v_max) and Acceleration:
  • v_max = (k/m)^{0.5} * A
  • a_max = (k/m) * A
  • Equations can be derived for specific displacements (x).

Mathematical Description of SHM

• Position Function: x = A cos(ωt)
  • ω = 2πf where f is the frequency.
• Velocity and Acceleration Functions:
  • v(t) = -v_max sin(ωt)
  • a(t) = -a_max cos(ωt)

Frequency and Period

• Period (T): Time to complete one cycle.
  • T = 1/f
• Frequency (f): Cycles per second (Hz).
  • f = 1/T
• Relations:
  • Period depends on mass (m) and spring constant (k): T = 2π (m/k)^{0.5}.
  • Higher mass increases period; higher spring constant decreases period.

Practical Problems and Calculations

• Various situations provided to calculate force, compression/stretch distances, energy, velocities, and spring constants.
• Examples include forces required to stretch/compress springs and system responses to changes in mass or spring constant.

Damped Harmonic Motion

• Types: Underdampened, overdampened, and critically damped systems.
• Graphical Representation:
  • Non-damped: Constant amplitude oscillations.
  • Damped: Reducing amplitude over time.
  • Critical Damping: Rapid return to equilibrium without overshooting.

Resonant Frequency

• Resonance occurs when the applied force frequency matches the system’s natural frequency, maximizing amplitude increase.
• Real-world example: Child on a swing.

Practice Problems Covered

• Force required to stretch a spring.
• Compression calculations.
• Effect of changing mass/spring constant on period and frequency.
• Calculating maximum acceleration, velocity, and potential energy.
• Examining periodic motion and energy transformations.
• Equations for SHM in terms of sine and cosine functions.
• Frequency of vibration calculations for different scenarios.

Summary: This lecture provided detailed information on SHM and periodic motion, focusing on the dynamics of a mass-spring system, energy transformations, and the mathematical equations governing these motions.