Oscillations and Equilibrium

• Oscillations: periodic back and forth motion
• Equilibrium Position: the position where the object would rest if all energy was removed from the system
• Equilibrium in Oscillations: always located at the center position (zero displacement)
• One side of the equilibrium position has a positive displacement, and the other side has a negative displacement

Time Period and Frequency

• Time Period (T): the amount of time taken for an oscillator to complete one full cycle, measured in seconds (s)
• Frequency (f): the number of cycles completed per second, measured in hertz (Hz)
• The time period and frequency are inversely related: T = 1/f

Angular Frequency

Angular Frequency (ω): the angle that an oscillator covers per second, measured in radians per second.

• One full cycle: 360° or 2π radians
• In one time period, an oscillator will complete one full cycle
• Hence:
• ω = 2π/T
• T = 2π/ω
• f = ω/2π

Simple Harmonic Motion

Simple Harmonic Motion: repeated motion around an equilibrium position where the acceleration of the object is proportional to its displacement, but in the opposite direction.

Acceleration vs Displacement Graphs

• Larger acceleration results in higher frequency and higher angular frequency
• Smaller acceleration results in lower frequency and lower angular frequency
• The gradient of lines give the negative square of the angular frequency: a = -ω²x

Simple Pendulums

• The time period is only affected by the length of the pendulum and the strength of gravity, and thus is independent of mass
• Acceleration of a simple pendulum: a = g⋅sin(θ)
• Period of a simple pendulum: T = 2π$\sqrt{\mathrm{L/g}}$, where L is the length of the pendulum (m), and g is the gravitational field strength (ms^-2)

Mass-Spring Oscillators

• The time period is affected by the mass and the spring constant
• However, the time period is independent of gravity and the amplitude
• Period of a mass-spring oscillator: T = 2π$\sqrt{\mathrm{m/k}}$, where m is the mass (kg), and k is the spring constant (N⋅m^-1)

Energy in Simple Harmonic Motion

• Pendulums: when it is at its peak (maximum displacement), all of its energy is gravitational potential energy, when it is at its equilibrium position, all of this energy is transferred to kinetic energy
• Horizontal Mass-Spring System: similar to pendulums, where the potential energy is elastic potential energy, but the same transfer to kinetic energy occurs
• Vertical Mass-Spring System: similar to horizontal mass-spring systems, where the potential energy is both gravitational potential energy and elastic potential energy at the top, and only elastic potential energy at the bottom, with part of this potential getting converted to kinetic energy at the equilibrium position
• In the absence of resistive forces (e.g. friction or air resistance), no energy is lost and oscillations continue indefinitely