Curso de Física. Tema 8: Oscilaciones. 8.1. Movimiento armónico simple

Introduction to Simple Harmonic Motion

Overview of Oscillations

• The topic begins with an introduction to oscillations, specifically focusing on simple harmonic motion (SHM), which is the simplest form of oscillatory movement.

• Oscillations and vibrations are prevalent in nature, observable in various biological systems such as mosquito wing movements and heartbeats, as well as in structures like bridges and buildings.

• A key characteristic of these phenomena is periodicity, where motion or displacement repeats at regular time intervals.

Characteristics of Periodic Motion

• Periodic motion can be simple or complex; for example, a basic graph shows a repeating pattern over time while more complex patterns can be seen in heartbeats.

• The goal is to study SHM by simplifying periodic motion to its most basic form, akin to the simpler graph presented earlier.

Understanding the Harmonic Oscillator

Initial Setup

• The discussion introduces a mass suspended from a spring at equilibrium. When displaced slightly from this position and released, it will begin to oscillate.

• This oscillation occurs between two points above and below the equilibrium position due to gravitational forces acting on the mass.

Effects of Damping

• While friction would eventually dampen this motion leading it towards equilibrium, without friction, the oscillation could theoretically continue indefinitely.

Relating SHM to Circular Motion

Visualizing Movement

• To simplify understanding SHM further, it’s compared with circular motion using a rotating disk with a pivot point that generates a shadow representing oscillation.

• As the disk rotates uniformly, the projection creates an oscillating pattern similar to that observed in SHM between two extreme positions.

Mathematical Representation

• The relationship between linear displacement (oscillation of mass on spring) and circular movement allows for deeper insights into SHM dynamics.

Equations for Simple Harmonic Motion

Position Equations

• To describe SHM mathematically, one can use coordinates based on circular motion: x = a cos(theta(t)) , where theta(t) changes over time reflecting angular velocity.

• Understanding how angle varies with time is crucial since it directly influences the position equations derived from circular motion principles.

Understanding Simple Harmonic Motion

Position Vector of Point P

• The position vector of point P is derived from the equation of uniform circular motion, expressed as angular velocity multiplied by time plus the initial angle.

• The components of the position vector are defined in terms of cosine and sine functions: x = A cos(Omega t + phi_0) and y = A sin(Omega t + phi_0) .

Projection on the X-Axis

• The focus shifts to the projection of point P on the x-axis rather than its circular motion, emphasizing that only this projection is relevant for analysis.

• The equation for this projection is given as x = A cos(Omega t + phi_0) , which is fundamental for understanding simple harmonic motion.

Equations and Their Significance

• Both cosine and sine functions can describe simple harmonic motion; however, cosine is typically preferred in equations. This flexibility allows for easier problem-solving depending on context.

• The oscillation occurs between maximum values defined by amplitude (A), with positions ranging from -A to A over time, centered around the origin.

Key Parameters in Simple Harmonic Motion

• Amplitude (A): Represents maximum displacement from equilibrium; it defines how far point P moves from its central position during oscillation. Values range between -A and A based on cosine function outputs.

• Angular Frequency ( Omega ): Relates to circular motion but specifically denotes frequency in simple harmonic contexts; it has similar units to angular velocity but applies differently here. It’s also referred to as pulsation in some texts.

Initial Phase Angle

• The term phi_0 within the cosine function represents the initial phase or angle at which oscillation begins, crucial for accurately describing motion at any given time. Understanding these terms will aid in solving related problems effectively.

Frequency Relationship

• Angular frequency ( Omega) can be calculated using 2pinu, where ν represents frequency; this relationship highlights how different parameters interconnect within harmonic systems, essential for deeper comprehension of oscillatory behavior.

Angular Frequency and Oscillations

Understanding Frequency

• Definition of Frequency: Frequency (denoted as Omega or nu) is defined as the number of complete oscillations per second. For example, if a point moves from position A to O and back to A, that constitutes one complete oscillation.

• Measurement Units: The frequency is measured in oscillations per second (Hz), which can also be expressed in seconds to the power of -1 (s⁻¹). Angular frequency relates to frequency through the formula omega = 2pi f .

Relationship Between Frequency and Period

• Concept of Period: The period is the time taken for one complete oscillation. It is inversely related to frequency; thus, T = 1/f and f = 1/T .

• Connecting Angular Frequency with Period: Just like we relate frequency with angular frequency, we can express angular frequency in terms of the period using derived expressions.

Measurement Units in Oscillatory Motion

• Distance Measurements: Both elongation (x) and amplitude (A) are distances measured in meters within the International System of Units.

• Phase Measurement: The initial phase angle is measured in radians. All calculations involving cosine must maintain radian measures for accuracy.

Calculating Angular Frequency

• Units Consistency: When calculating angular displacement ( Omega t ), ensure units remain consistent; multiplying radian/second by seconds yields radians.

• Expression for Angular Frequency: An important expression for angular frequency based on mass and spring constant will be introduced later.

Formulae Related to Simple Harmonic Motion

• Frequency Calculation Formula: The angular frequency can be calculated using omega = sqrtk/m , where k is the spring constant and m is mass.

• Position Formula Overview: The main formula governing simple harmonic motion will outline how various parameters interact throughout this topic.