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