Simple Harmonic Motion Concepts

Simple Harmonic Motion

• The acceleration of an object in simple harmonic motion (SHM) is proportional to its displacement from the equilibrium position, and always acts towards the equilibrium position.
• The key equation is a=−ω2xa=-ω^2xa=−ω2x
• Where:
  • aaa is the acceleration
  • ωωω is the angular frequency
  • xxx is the displacement
• The minus sign indicates that the acceleration is always in the opposite direction to the displacement.
• The period of an oscillator in SHM is independent of its amplitude, this is called isochronous oscillation
• The period (TTT) is the time taken to complete one full oscillation.
• The frequency (fff) is the number of oscillations per second, and is the reciprocal of the period (1/T).
• The phase difference (ϕ) is how much of an oscillation one oscillator is "ahead" or "behind" another.
• Angular frequency (ω) is the rate of change of angular position. It is related to the period by the equation: ω=2πf=2π/Tω=2πf=2π/Tω=2πf=2π/T
• Displacement can be calculated using:
  • x=Asin⁡(ωt)x = A\sin(ωt)x=Asin(ωt) if the oscillator starts at the equilibrium position
  • x=Acos⁡(ωt)x = A\cos(ωt)x=Acos(ωt) if the oscillator starts at the amplitude position
  • Where:
    • AAA is the amplitude of the oscillation
    • ttt is the time
• The velocity of the oscillator is given using the equation: v=±ω√(A2−x2)v=±ω√(A^2−x^2)v=±ω√(A2−x2)
• The maximum velocity occurs at the equilibrium position, where x=0x=0x=0. The maximum velocity is vmax=ωAv_{max}=ωAvmax​=ωA
• The maximum acceleration occurs at the amplitude points, and is 0 at the equilibrium position.

Energy Transfers in SHM

• Energy is exchanged between kinetic and potential forms during SHM.
• The maximum kinetic energy occurs at the equilibrium point, where the velocity is at a maximum.
• The maximum potential energy occurs at the amplitude positions, where displacement is at a maximum.
• The total energy of the oscillator is always conserved.

Damping

• Damping refers to the reduction in the amplitude of oscillations over time.
• This reduction is caused by energy loss due to resistive forces like friction or drag.
• Light damping occurs naturally (like a pendulum oscillating in air), and the amplitude decreases exponentially.
• Heavy damping occurs when the oscillations are significantly reduced by a resistive force (like a pendulum oscillating in water). The amplitude decreases dramatically.
• Critical damping occurs when the object stops before completing one oscillation. This often happens when the object is submerged in a very viscous fluid (like treacle).

Resonance

• A free oscillation occurs when an object oscillates without external forces being applied, at its natural frequency.
• A forced oscillation occurs when an object is subjected to a periodic external driving force, making it oscillate at that specific frequency.
• Resonance occurs when the driving frequency of the external force matches the natural frequency of the object.
• The amplitude of oscillations increases rapidly at resonance, and can continue to increase until the system fails if there is no damping.
• Increasing damping reduces the amplitude at all frequencies, and shifts the peak amplitude to a lower frequency.

Investigating Resonance

• Resonance can be investigated experimentally by using a mass suspended between two springs attached to an oscillation generator.
• The driver frequency of the generator is slowly increased from zero.
• The amplitude of oscillation increases as the driver frequency approaches the natural frequency of the system, reaching a maximum at resonance.
• Then, with increasing frequency, the amplitude decreases again.
• The spring-mass system experiences damping from the air, so the amplitude will not continue to increase indefinitely.
• To increase accuracy, the system can be filmed for later analysis.