OSCILLATIONS (AP Physics 1: Algebra based)
Simple Harmonic Motion (SHM)
Definition
Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction.
Mathematically: F = −kx
F: Restoring force
k: Spring constant (N/m)
x: Displacement from equilibrium (m)
The motion is sinusoidal in nature, leading to predictable oscillations
Common systems exhibiting SHM:
Mass-spring systems
Simple pendulums (for small angles)
Tuning forks
Vibrating strings
Frequency and Period of SHM
Definitions
Period (T): Time taken for one complete cycle of motion.
Units: seconds (s)
Frequency (f): Number of cycles per unit time.
Units: hertz (Hz)
Relationship: f = 1/T
Formulas
Mass-Spring System (Horizontal or Vertical)
Period: T = 2π √m/k
m: Mass attached to the spring (kg)
k: Spring constant (N/m)
Simple Pendulum (Small Angle Approximation)
Period: T = 2π √L/g
L: Length of the pendulum (m)
g: Acceleration due to gravity (9.81 m/s²)
Key Points
The period of SHM is independent of amplitude (for ideal systems).
In a mass-spring system, increasing mass mmm increases the period T.
In a pendulum, increasing length L increases the period T.
Representing and Analyzing SHM
🔹 Displacement, Velocity, and Acceleration
Displacement: x(t) = A cos(ωt+ϕ)
Velocity: v(t) = −Aω sin(ωt+ϕ)
Acceleration: a(t) = −Aω²cos(ωt+ϕ)
A: Amplitude (maximum displacement)
ω: Angular frequency = 2π f
ϕ: Phase constant (depends on initial conditions)
Graphical Representations
Displacement-Time Graph: Cosine wave starting at maximum displacement.
Velocity-Time Graph: Sine wave, 90° out of phase with displacement.
Acceleration-Time Graph: Cosine wave, 180° out of phase with displacement.
Phase Relationships
Velocity is 90° out of phase with displacement.
Acceleration is 180° out of phase with displacement
Energy in Simple Harmonic Oscillators
Energy Types
Kinetic Energy (KE): Energy due to motion.
KE = ½ mv²
Potential Energy (PE): Energy stored in the system.
For springs: PE = ½ kx²
Total Mechanical Energy
In ideal SHM (no damping), total energy EEE remains constant:
E = KE + PE = ½ kA²
Energy Transformations
At maximum displacement (x = ±A):
KE = 0
PE = Maximum
At equilibrium position (x = 0):
KE = Maximum
PE = 0
Energy Graphs
KE and PE vary sinusoidally over time, with total energy remaining constant.
Examples of SHM
Mass-Spring System
A mass attached to a spring oscillates horizontally or vertically.
The restoring force is provided by the spring's elasticity.
Simple Pendulum
A mass (bob) suspended from a string swings back and forth.
For small angles (<15°), the motion approximates SHM.
Other Examples
Vibrating guitar strings.
Tuning forks.
Air columns in wind instruments.
Tips for AP Physics 1 Exam
Understand the derivations of period formulas for different systems.
Be able to analyze energy transformations within SHM.
Practice interpreting and sketching displacement, velocity, and acceleration graphs
Apply SHM concepts to real-world systems and identify approximations made