Simple harmonic motion is the backbone of oscillations and waves. It's everywhere, from pendulums swinging to springs bouncing. Understanding its key features like period, frequency, and amplitude helps us grasp how these motions work.
Energy plays a big role in harmonic motion, constantly shifting between kinetic and potential forms. Phase shifts and mass-spring systems add complexity, showing how these motions can interact and be described mathematically. It's all about patterns and predictability in motion.
Simple Harmonic Motion
Period and frequency in harmonic motion
- Period ($T$) time required for an oscillating object to complete one full cycle of motion
- Measured in seconds (s)
- Calculated using the equation $T = \frac{1}{f}$, where $f$ is the frequency
- Examples: pendulum swing, vibrating string
- Frequency ($f$) number of oscillations or cycles per unit time
- Measured in hertz (Hz) or cycles per second (s$^{-1}$)
- Calculated using the equation $f = \frac{1}{T}$, where $T$ is the period
- Examples: tuning fork vibration, alternating current
Characteristics of harmonic motion
- Restoring force directly proportional to displacement and acts in opposite direction
- Described by the equation $F = -kx$, where $k$ is the spring constant and $x$ is the displacement from equilibrium
- Examples: spring force, pendulum gravitational force
- Motion sinusoidal, with object oscillating back and forth about equilibrium position
- Follows a smooth, repetitive pattern
- Examples: wave motion, vibrating string
- Amplitude ($A$) maximum displacement from equilibrium position
- Determines the energy of the oscillation
- Examples: wave height, pendulum swing angle
- Motion periodic, repeating itself at regular time intervals (period)
- Allows for predictable behavior
- Examples: clock pendulum, AC voltage
- Total energy (kinetic + potential) remains constant throughout motion
- Energy converts between kinetic and potential forms
- Examples: spring-mass system, pendulum swing
Phase shift in oscillations
- Phase shift ($\phi$) difference in starting point of oscillation between two or more simple harmonic motions
- Measured in radians or degrees
- Represents a temporal offset between oscillations
- Examples: sound wave interference, coupled pendulums
- Phase shift caused by differences in initial conditions or external factors
- Initial position, velocity, or applied force can affect phase
- Examples: two pendulums released at different times, driven oscillations
- Oscillations with same frequency and amplitude but different phase shifts have maximum and minimum displacements occurring at different times
- Results in constructive or destructive interference when combined
- Examples: sound waves from multiple sources, light wave interference
- Displacement with phase shift described by the equation $x(t) = A \cos(\omega t + \phi)$
- $\omega$ is the angular frequency
- $\phi$ determines the initial displacement and direction of motion
- Examples: shifted cosine wave, coupled harmonic oscillators
- Angular displacement (θ) represents the angle between the oscillating object and its equilibrium position
- Relates to linear displacement in circular motion and pendulums
Equations for mass-spring systems
- Equation of motion for mass-spring system derived using Newton's second law ($F = ma$) and Hooke's law ($F = -kx$)
- $ma = -kx$
- $m\frac{d^2x}{dt^2} = -kx$, where $m$ is mass and $x$ is displacement
- Examples: spring-mass oscillator, vibrating molecule
- Solution to differential equation for displacement:
- $x(t) = A \cos(\omega t + \phi)$
- $\omega = \sqrt{\frac{k}{m}}$ is the angular frequency
- Examples: position of mass on spring, displacement of vibrating string
- Velocity found by differentiating position equation
- $v(t) = -A\omega \sin(\omega t + \phi)$
- Represents the rate of change of displacement
- Examples: speed of mass on spring, velocity of vibrating particle
- Acceleration found by differentiating velocity equation or using original equation of motion
- $a(t) = -A\omega^2 \cos(\omega t + \phi)$
- Represents the rate of change of velocity
- Examples: acceleration of mass on spring, acceleration of vibrating object
Energy of vertical spring oscillations
- Mass oscillating on vertical spring experiences restoring force and gravity
- Equilibrium position slightly lower than relaxed spring length due to mass weight
- Examples: spring-mass system hanging vertically, bungee jumper
- Effective spring constant ($k_{eff}$) same as original spring constant ($k$)
- Gravitational force does not affect the stiffness of the spring
- Examples: spring-mass system in vertical orientation, suspension bridge
- Equation of motion same as horizontal mass-spring system, with shifted equilibrium position
- Gravitational force accounted for in the new equilibrium position
- Examples: vertical spring-mass oscillator, diving board
- Potential energy includes elastic potential ($\frac{1}{2}kx^2$) and gravitational potential ($mgy$)
- Elastic potential energy stored in the compressed or stretched spring
- Gravitational potential energy depends on the height of the mass relative to a reference point
- Examples: compressed spring, object raised above ground
- Kinetic energy ($\frac{1}{2}mv^2$) same as in horizontal case
- Depends on the velocity of the oscillating mass
- Examples: moving mass on spring, bouncing object
- Total energy (kinetic + elastic potential + gravitational potential) remains constant throughout motion
- Energy converts between forms as mass oscillates
- Examples: energy conservation in spring-mass system, energy transfer in bouncing ball
Advanced oscillation concepts
- Resonance occurs when an oscillating system is driven at its natural frequency, resulting in maximum amplitude
- Natural frequency is the frequency at which a system oscillates freely without external forces
- Damping reduces the amplitude of oscillations over time due to energy dissipation
- Examples: shock absorbers, pendulum motion in air
- Forced oscillation occurs when an external periodic force is applied to an oscillating system
- Can lead to resonance if the driving frequency matches the system's natural frequency