Periodic Motion

Introduction to Periodic Motion

Periodic motion refers to any motion that repeats itself in a regular cycle over time. This type of motion is fundamental in physics and is observed in systems such as pendulums, vibrating strings, and oscillating springs.

• Definition: Motion that repeats identically after a fixed time interval, called the period (T).

• Examples: Vibrations of machines, musical instruments, pendula, and planetary orbits.

• Cause: A restoring force that acts to return the system to equilibrium, combined with inertia.

Parameters Describing Oscillatory Motion

• Period (T): The time for one complete cycle (measured in seconds).

• Frequency (f): The number of cycles per second (measured in hertz, Hz).

• Angular Frequency (\omega): (measured in radians per second).

• Equilibrium Position: The position where the net force is zero.

• Displacement (x): Distance from equilibrium position.

• Amplitude (A): Maximum displacement from equilibrium.

• Total Mechanical Energy: For undamped systems, , where is kinetic energy and is potential energy.

Simple Harmonic Motion (SHM)

Definition and Characteristics

Simple harmonic motion is a specific type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.

• Restoring Force: (Hooke's Law for springs)

• Displacement Equation: or

• Amplitude (A): Maximum value of displacement.

• Phase (\varphi): Determines the initial position and direction of motion.

Equations of Motion for SHO

• Newton's Second Law:

• General Solution:

• Angular Frequency:

• Period:

• Frequency:

Initial Conditions and Phase

• Initial Position and Velocity: Determine amplitude and phase.

• For zero initial phase:

• Velocity:

• Acceleration:

Energy in Simple Harmonic Motion

• Kinetic Energy:

• Potential Energy:

• Total Mechanical Energy:

• Energy Conversion: Energy oscillates between kinetic and potential forms as the mass moves.

Example: Mass on a Spring

• Given:

• Frequency:

• First at cm: s

Relation of Uniform Circular Motion to SHO

Simple harmonic motion can be visualized as the projection of uniform circular motion onto one axis.

• Phasor Representation: A vector rotating with constant angular velocity .

• Projection: , where is the radius of the circle.

Pendulums and Physical Oscillators

The Simple Pendulum

• Definition: A point mass suspended by a weightless string.

• Restoring Force: (for small angles)

• Period: (independent of mass, valid for small angles)

The Physical Pendulum

• Definition: Any rigid body oscillating about a pivot point.

• Restoring Torque:

• For small angles:

• Period: , where is the moment of inertia about the pivot.

Torsion Pendulum

• Restoring Torque: , where is the torsion constant.

• Angular Frequency:

• Period:

Damped and Driven Oscillations

Damped Oscillations

Real-world oscillators experience friction or resistance, causing the amplitude to decrease over time (damping).

• Damping Force: (proportional to velocity)

• Displacement with Damping: , where

• Regimes: Underdamped (), Critically damped (), Overdamped ()

Forced Oscillations and Resonance

• Forced Oscillation: Occurs when an external periodic force drives the system.

• Resonance: When the driving frequency matches the natural frequency, the amplitude becomes maximal.

• Applications: Shock absorbers, musical instruments, bridge oscillations (e.g., Tacoma Narrows Bridge).

Key Terminology

• Restoring Force

• Simple Harmonic Motion (SHM)

• Hooke's Law

• Amplitude (A)

• Cycle

• Oscillation

• Period (T)

• Frequency (f)

• Angular Frequency (\omega)

Summary Table: Key Equations in Simple Harmonic Motion

Applications and Examples

• Mass-Spring System: Used to model many mechanical oscillators.

• Pendulums: Used in clocks and timekeeping devices.

• Molecular Vibrations: Atoms in molecules oscillate about equilibrium positions, modeled as SHM for small displacements.

• Resonance Phenomena: Engineering structures must avoid resonance frequencies to prevent catastrophic failure.