Introduction to Oscillations

Introduction

In the study of mechanics, we have so far mainly considered translational motion (motion along a straight line or curved path) and rotational motion. Now, we turn our attention to a special and very common type of motion: oscillatory motion.

Oscillation (or vibration) refers to a repetitive motion where an object or a system moves back and forth about a central, stable equilibrium position. Many phenomena in physics and everyday life involve oscillations.

Examples of oscillatory motion include:

• The swinging of a pendulum.
• The vibrations of a string on a musical instrument.
• The up-and-down motion of a mass attached to a spring.
• The vibrations of atoms in a crystal lattice.
• The oscillations of a bridge or building under wind or seismic forces.
• The oscillation of the balance wheel in a mechanical watch.

Oscillatory motion is closely related to wave phenomena, as waves are often described as the propagation of oscillations through a medium or space.

Understanding oscillations requires defining new terms like period, frequency, and amplitude, and exploring the conditions under which oscillations occur. A key characteristic of many oscillations is the presence of a restoring force that always acts to bring the object back towards its equilibrium position, and inertia, which causes it to overshoot the equilibrium, leading to the back-and-forth motion.

The simplest and most fundamental type of oscillation is called Simple Harmonic Motion (SHM), which we will study in detail. Many complex oscillations can be analysed as a combination of simple harmonic motions.

Periodic And Oscillatory Motions

To describe and analyse oscillatory motion, we first need to understand the broader category of periodic motion and the specific characteristics that define an oscillation.

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Periodic Motion

A motion is said to be periodic if it repeats itself at regular intervals of time. In periodic motion, the position and velocity of the object at any given instant are the same as they were at specific earlier instants, separated by a fixed time interval.

Examples of periodic motion include:

• The motion of the hands of a clock.
• The motion of a planet around the Sun (approximately periodic).
• The rotation of the Earth about its axis.
• The motion of a swing (if friction is ignored).

Periodic motion does not necessarily involve back-and-forth movement about a single point. For example, circular motion at a constant speed is periodic but not oscillatory.

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Oscillatory Motion

Oscillatory motion is a type of periodic motion in which the object moves back and forth about a fixed point or position called the equilibrium position. The equilibrium position is usually a position of stable equilibrium, where the net force on the object is zero, and if displaced slightly, a restoring force acts to bring it back towards equilibrium.

All oscillatory motions are periodic, but not all periodic motions are oscillatory.

Examples of oscillatory motion:

• The motion of a pendulum bob.
• A mass vibrating on a spring.
• A tuning fork vibrating.

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Harmonic Motion

When the motion can be described by sinusoidal functions (like sine or cosine), it is called harmonic motion. Simple Harmonic Motion is the simplest form of harmonic motion.

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Period And Frequency ($ T = 1/\nu $)

For any periodic motion, including oscillatory motion, two important characteristics are its period and frequency.

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Period (T)

The period ($T$) of a periodic motion is the minimum time interval after which the motion repeats itself. It is the time taken to complete one full cycle of the motion.

Units of period are typically seconds (s).

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Frequency ($\nu$ or f)

The frequency ($\nu$ or $f$) of a periodic motion is the number of cycles completed per unit time. It is the reciprocal of the period.

$ \nu = \frac{1}{T} $

The SI unit of frequency is Hertz (Hz), where $1 \text{ Hz} = 1$ cycle per second. Other units like cycles per minute (cpm) or revolutions per minute (rpm) are also used in different contexts.

Related to frequency is angular frequency ($\omega$), often used in describing harmonic motion. Angular frequency is the rate of change of phase angle (in radians) per unit time. For periodic motion with period $T$ and frequency $\nu$, the angular frequency is:

$ \omega = \frac{2\pi}{T} = 2\pi\nu $

Units of angular frequency are radians per second (rad/s).

Answer:

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Displacement

In oscillatory motion, the position of the oscillating object is constantly changing relative to its equilibrium position. This position relative to the equilibrium position is often referred to as the displacement.

In linear oscillations (like a mass on a spring moving back and forth along a line), the displacement ($x$) is typically measured as the distance from the equilibrium position. The equilibrium position is usually taken as $x=0$. The displacement can be positive or negative depending on which side of the equilibrium position the object is.

In angular oscillations (like a pendulum swinging), the displacement can be described by the angular position ($\theta$) relative to the equilibrium position (usually the vertical position for a simple pendulum). The equilibrium angular position is usually taken as $\theta=0$.

The maximum displacement from the equilibrium position is called the amplitude ($A$ or $\theta_0$). The amplitude is always a positive value. The actual displacement $x$ (or $\theta$) varies periodically between $-A$ and $+A$.

For simple harmonic motion, the displacement as a function of time is typically described by a sinusoidal function:

$ x(t) = A \cos(\omega t + \phi) $

or

$ x(t) = A \sin(\omega t + \phi) $

where:

• $A$ is the amplitude.
• $\omega$ is the angular frequency.
• $t$ is time.
• $\phi$ is the initial phase constant, determined by the position and velocity at $t=0$.

The displacement is a crucial variable in describing the instantaneous state of an oscillating system.