Glossary
Acceleration (in SHM)
The rate of change of velocity with respect to time for an oscillating object. In SHM, it is sinusoidal and directly proportional to the negative of the displacement.
Example:
At the extreme ends of a swing's motion, its acceleration is at its maximum, pulling it back towards the center.
Amplitude (A)
The maximum displacement or distance moved by an oscillating object from its equilibrium position.
Example:
If a swing goes 2 meters forward from its lowest point, its amplitude is 2 meters.
Angular Frequency (ω)
A measure of the rate of oscillation, representing the number of radians per second an object completes in its cycle. It is directly related to the period and frequency of the motion.
Example:
A speaker cone vibrating at 440 Hz has a specific angular frequency that determines the pitch of the sound.
Displacement (in SHM)
The position of an oscillating object relative to its equilibrium position at any given time, typically described by a sinusoidal function.
Example:
If a pendulum swings 10 cm to the right from its center, its displacement is +10 cm.
Frequency (f)
The number of complete cycles of motion that occur per unit of time. It is the inverse of the period.
Example:
A guitar string vibrating 440 times per second has a frequency of 440 Hz.
Generic Differential Equation of SHM
A second-order differential equation, $\frac{d^2x}{dt^2} = -\omega^2 x$, that universally describes any system undergoing Simple Harmonic Motion.
Example:
Deriving this Generic Differential Equation of SHM from Newton's second law for a mass-spring system confirms it exhibits SHM.
Kinetic Energy (K)
The energy an object possesses due to its motion, calculated as $K = \frac{1}{2}mv^2$.
Example:
A roller coaster car at the bottom of a dip has maximum kinetic energy due to its high speed.
Period (T)
The time it takes for one complete cycle of periodic motion to occur.
Example:
If a clock's pendulum completes one full swing (back and forth) in 1 second, its period is 1 second.
Periodic Motion
Any motion that repeats itself over a consistent time interval. SHM is a specific type of periodic motion.
Example:
The Earth's orbit around the Sun is a form of periodic motion, repeating approximately every 365 days.
Phase Angle (φ)
An initial angle that determines the starting position of an oscillating object at time t=0 within its cycle, influencing the specific sinusoidal function used.
Example:
If a mass on a spring starts at its maximum positive displacement, its initial phase angle would be zero if using a cosine function.
Potential Energy (Gravity) (U_g)
The energy an object possesses due to its position in a gravitational field, calculated as $U_g = mgh$.
Example:
A book on a high shelf has more potential energy (gravity) than the same book on a lower shelf.
Potential Energy (Spring) (U_s)
The energy stored in a spring when it is stretched or compressed from its equilibrium position, calculated as $U_s = \frac{1}{2}kx^2$.
Example:
A stretched slingshot holds potential energy (spring), ready to be converted into kinetic energy.
Restoring Force
A force that always acts to bring an object back to its equilibrium position. In SHM, this force is directly proportional to the displacement from equilibrium and acts in the opposite direction.
Example:
When you stretch a rubber band, the elastic force pulling it back to its original shape is a restoring force.
Simple Harmonic Motion (SHM)
A special type of periodic motion where an object oscillates back and forth around a stable equilibrium position, characterized by a restoring force proportional to the displacement.
Example:
A mass bouncing on a spring demonstrates Simple Harmonic Motion, moving rhythmically up and down around its resting point.
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 stiff car suspension spring would have a high spring constant, requiring a large force to compress it.
Total Mechanical Energy (ME)
The sum of an object's kinetic and potential energies. In an ideal SHM system without non-conservative forces, this energy remains constant.
Example:
In a frictionless pendulum swing, the total mechanical energy remains constant, continuously converting between kinetic and gravitational potential energy.
Velocity (in SHM)
The rate of change of displacement with respect to time for an oscillating object. In SHM, it is also sinusoidal and reaches its maximum magnitude at the equilibrium position.
Example:
As a bungee jumper passes the lowest point of their jump, their velocity momentarily becomes zero before they spring back up.