Glossary
Acceleration
The rate of change of an object's velocity in SHM, always directed towards the equilibrium position and proportional to the negative of the displacement.
Example:
When a bungee jumper reaches the lowest point of their jump, their upward acceleration is at its maximum as the cord stretches to its fullest.
Acceleration (in SHM)
The rate of change of velocity in SHM, which is always proportional to the displacement and directed opposite to it, given by $a = -\omega^2 x$.
Example:
At the extreme ends of a spring's oscillation, the acceleration is maximum and points towards the equilibrium.
Acceleration-Time Graph
A graph that plots the acceleration of an object undergoing SHM as a function of time, appearing sinusoidal and shifted by half a period relative to displacement.
Example:
The acceleration-time graph for a simple pendulum shows maximum acceleration when the pendulum is at its extreme swing points.
Amplitude
The maximum displacement or distance moved by an oscillating object from its equilibrium position.
Example:
If a pendulum swings 10 cm from its center point to its highest point, its amplitude is 10 cm.
Amplitude (A)
The maximum displacement or distance moved by a point on a vibrating body or wave measured from its equilibrium position.
Example:
If a guitar string vibrates 2 mm from its resting position, its amplitude is 2 mm.
Angular Frequency ($\omega$)
A measure of the rate of oscillation, expressed in radians per second, and related to frequency (f) by $\omega = 2\pi f$.
Example:
A spring-mass system with an angular frequency of 10 rad/s completes oscillations faster than one with 5 rad/s.
Differential Equation for SHM
A second-order differential equation, $\frac{d^2x}{dt^2} = -\omega^2 x$, that mathematically describes the motion of an object undergoing Simple Harmonic Motion.
Example:
Recognizing that a system's motion satisfies the Differential Equation for SHM immediately tells you its oscillations are sinusoidal.
Displacement
The position of an oscillating object relative to its equilibrium position at any given time.
Example:
If a pendulum bob is pulled 5 cm to the right from its lowest point, its displacement is +5 cm.
Displacement (in SHM)
The instantaneous position of an object in SHM relative to its equilibrium position, described by a sinusoidal function of time.
Example:
When a block on a spring is pulled 5 cm to the right from its resting point, its displacement is +5 cm.
Displacement-Time Graph
A graph that plots the position of an object undergoing SHM as a function of time, typically showing a sinusoidal curve.
Example:
A displacement-time graph for a bouncing ball would show its height changing sinusoidally over time if it were undergoing ideal SHM.
Equilibrium Position
The stable position where the net force on an oscillating object is zero, and it would remain at rest if undisturbed.
Example:
For a mass hanging from a spring, the equilibrium position is where the spring's upward force perfectly balances the mass's weight.
Frequency
The number of complete oscillations or cycles that occur per unit of time, typically measured in Hertz (Hz).
Example:
If a spring completes 5 oscillations in 1 second, its frequency is 5 Hz.
Frequency (f)
The number of complete oscillations or cycles that occur per unit of time, typically measured in Hertz (Hz).
Example:
If a tuning fork completes 440 vibrations in one second, its frequency is 440 Hz.
Graphical Analysis
The process of interpreting and extracting information about SHM by examining displacement-time, velocity-time, and acceleration-time graphs.
Example:
By performing graphical analysis of a seismograph's output, scientists can determine the amplitude and frequency of earthquake waves.
Graphical Analysis of SHM
The interpretation of graphs (displacement-time, velocity-time, acceleration-time) to understand the phase relationships and characteristics of Simple Harmonic Motion.
Example:
By performing Graphical Analysis of SHM, one can observe that the velocity graph is shifted 90 degrees relative to the displacement graph.
Length (L) of Pendulum
The distance from the pivot point to the center of mass of the pendulum bob in a simple pendulum.
Example:
To make a pendulum clock tick slower, you would need to increase the length (L) of the pendulum.
Mass-Spring System
A common physical model for SHM consisting of a mass attached to a spring, oscillating horizontally or vertically.
Example:
A toy car attached to a spring that bounces back and forth on a frictionless track demonstrates a mass-spring system.
Maximum Acceleration ($a_{max}$)
The greatest magnitude of acceleration experienced by an object in SHM, occurring at the points of maximum displacement (the extremes of oscillation).
Example:
When a mass on a spring momentarily stops at its furthest stretched point, it experiences its maximum acceleration back towards equilibrium.
Maximum Velocity ($v_{max}$)
The greatest speed attained by an object undergoing SHM, which occurs when the object passes through its equilibrium position.
Example:
A pendulum swings fastest at the bottom of its arc, where its maximum velocity is reached.
Natural Frequency
The specific frequency at which a system will oscillate freely when disturbed from its equilibrium position and then left to oscillate without any external driving forces.
Example:
Every musical instrument has a natural frequency at which it vibrates most easily to produce sound.
Period (T)
The time it takes for an object undergoing SHM to complete one full oscillation or cycle.
Example:
If a grandfather clock's pendulum takes 2 seconds to swing back and forth once, its period is 2 seconds.
Period (of SHM)
The time taken for one complete oscillation or cycle of motion in Simple Harmonic Motion.
Example:
If a mass on a spring takes 0.5 seconds to complete one full back-and-forth motion, its period is 0.5 s.
Phase Constant ($\phi$)
An angle in the sinusoidal equation of SHM that determines the initial position or state of the oscillating object at time t=0.
Example:
If an object starts at its maximum positive displacement at t=0, its phase constant is 0.
Phase Relationships
The relative timing or shift between the sinusoidal curves of displacement, velocity, and acceleration in SHM.
Example:
Understanding the phase relationships helps predict that when displacement is maximum, velocity is zero.
Resonance
A phenomenon where the amplitude of oscillation of a system increases dramatically when an external driving force is applied at or near the system's natural frequency.
Example:
Pushing a swing at its natural rhythm causes its amplitude to grow due to resonance.
Simple Harmonic Motion (SHM)
A type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction, causing an object to oscillate back and forth around an equilibrium point.
Example:
A simple harmonic motion is observed when a child swings on a playground swing, moving back and forth in a regular, repeating pattern.
Simple Harmonic Motion (SHM)
A type of oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction, resulting in a sinusoidal oscillation.
Example:
A mass attached to a spring oscillating back and forth on a frictionless surface demonstrates Simple Harmonic Motion.
Simple Pendulum
An idealized model of SHM consisting of a point mass suspended by a massless, inextensible string, swinging freely under gravity.
Example:
A child's swing, when swinging with small angles, approximates a simple pendulum.
Spring Constant (k)
A measure of the stiffness of a spring, representing the force required to stretch or compress the spring by a unit distance.
Example:
A very stiff car suspension spring would have a high spring constant, requiring a large force to compress it.
Velocity
The rate of change of an object's displacement in SHM, indicating both its speed and direction of motion.
Example:
As a mass on a spring passes through its equilibrium point, its velocity reaches its maximum magnitude.
Velocity-Time Graph
A graph that plots the velocity of an object undergoing SHM as a function of time, appearing sinusoidal and shifted by a quarter period relative to displacement.
Example:
On a velocity-time graph for a mass on a spring, the curve would be at its maximum when the displacement is zero.