Vibrations and Waves Summary

Periodic Motion: Events that repeat over time. Examples include:
- Playground swings
- Pendulums
- Wrecking balls
Simple Harmonic Motion (SHM): A specific type of periodic motion where the restoring force is directly proportional to displacement.
Hooke's Law: Describes how the spring force (F) relates to displacement (x) as follows: $F_{elastic} = -kx$
- Where:
- k = spring constant (stiffness of the spring)
- x = displacement from equilibrium position
- The negative sign indicates the force acts in the opposite direction of displacement.

Equilibrium Position (x=0): At this point, force and acceleration are zero, but speed is maximum.
Maximum Displacement: At maximum displacement, both spring force and acceleration are maximal, while speed is zero.
Damping: In real systems, motion decreases over time due to friction (not ideal).
Restoring Force: It is always directed toward the equilibrium position and proportional to displacement.

Elastic Potential Energy: Energy stored in the spring when stretched or compressed.
Mechanical Energy Conservation: In a frictionless environment, total mechanical energy (potential + kinetic) remains constant.

Definition: A mass (bob) attached to a string (or rod) swinging back and forth.
The restoring force is the component of gravitational force acting along the path of motion (tangential force).
SHM Conditions: The motion is SHM for small angles (less than 15 degrees), where displacement is proportional to the angle.
Energy Conversion: Potential energy (maximum at the peaks) converts into kinetic energy (maximum at the lowest point).

Amplitude (A): The maximum distance from the equilibrium position.
Period (T): The time taken to complete one full cycle.
Frequency (f): The number of cycles per second; inversely related to period:
$f = \frac{1}{T}$ .

For a Pendulum:
$T = 2π \sqrt{\frac{L}{g}}$
Where L is the length of the pendulum, and g is the acceleration due to gravity.
For a Mass-Spring System:
$T = 2π \sqrt{\frac{m}{k}}$
Where m is mass and k is the spring constant.

Wave Definition: A disturbance that transfers energy through a medium without transferring matter.
Medium: The substance through which a wave travels (e.g., air, water).
Types of Waves:
- Transverse Waves: Particles move perpendicular to wave direction (e.g., waves on a string).
- Longitudinal Waves: Particles move parallel to wave direction (e.g., sound waves in air).
Frequency and Wavelength Relationship:
$v = f λ$
Where v is wave speed, f is frequency, and λ is wavelength.

Superposition Principle: When two waves overlap, their effects add together point by point.
Constructive Interference: Occurs when waves add up to create a larger amplitude.
Destructive Interference: Occurs when waves cancel each other out, resulting in smaller or zero amplitude.
Standing Waves: Results from the interference of two waves traveling in opposite directions at the same frequency. Characterized by fixed points (nodes) and points of maximum movement (antinodes).

What characterizes SHM?: Restoring force proportional to displacement.
What examples exist for SHM?: Springs, pendulums, etc.
What happens to acceleration in SHM?: It constantly changes.
How does the period of a pendulum depend on its length?: Doubling the length increases the period.

For a Pendulum:

$T = 2π \sqrt{\frac{L}{g}}$
Where:

For a Mass-Spring System:

$T = 2π \sqrt{\frac{m}{k}}$
Where:

Frequency and Wavelength Relationship:

$v = f \lambda$
Where:
v = wave speed
f = frequency
λ = wavelength

For a Pendulum:

$T = 2π \sqrt{\frac{L}{g}}$
Where:

For a Mass-Spring System:

$T = 2π \sqrt{\frac{m}{k}}$
Where:

Frequency and Wavelength Relationship:

$v = f \lambda$
Where:
v = wave speed
f = frequency
λ = wavelength