Introduction

Motion can be categorized in many ways. We have studied non-repetitive motions like rectilinear motion and projectile motion. We have also studied repetitive, or periodic motion, such as uniform circular motion. This chapter focuses on a specific type of periodic motion known as oscillatory motion, where an object moves to and fro about a mean position. Examples include a child on a swing, the pendulum of a clock, and a boat bobbing on water.

The study of oscillations is fundamental to physics as it helps in understanding many phenomena. Vibrating strings in musical instruments, diaphragms in speakers, and the vibrations of atoms in a solid are all examples of oscillatory motion. These phenomena are described using fundamental concepts like period, frequency, displacement, amplitude, and phase, which will be developed in this chapter.

Periodic and Oscillatory Motions

A motion that repeats itself at regular intervals of time is called periodic motion. If a body in periodic motion moves back and forth about a stable equilibrium position, the motion is called oscillatory or vibratory. All oscillatory motions are periodic, but not all periodic motions are oscillatory (e.g., uniform circular motion).

An oscillatory motion is typically caused by a restoring force. When a body is displaced from its equilibrium position, a net force acts on it, trying to bring it back to that equilibrium point. This force gives rise to the oscillations.

The simplest and most important type of oscillatory motion is simple harmonic motion (SHM), which occurs when the restoring force is directly proportional to the displacement from the mean position and is always directed towards it.

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Period, Frequency, and Displacement

Period (T) and Frequency ($\nu$)

• The period (T) is the smallest interval of time after which the motion repeats itself. Its SI unit is the second (s).
• The frequency ($\nu$) is the number of repetitions (or cycles) per unit time. It is the reciprocal of the period.

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

The SI unit of frequency is s⁻¹, which is named the hertz (Hz). 1 Hz = 1 cycle per second.

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Displacement

In the context of oscillations, displacement refers to the change in any physical property of the system as a function of time. While it often means position (e.g., the distance of a block from its equilibrium position), it can also refer to other variables like the angle of a pendulum, the voltage across a capacitor in an AC circuit, or the pressure variation in a sound wave.

The displacement in a periodic motion can be represented by a periodic function of time. The simplest periodic functions are sine and cosine functions, for example:

$f(t) = A \cos(\omega t)$

This function repeats itself when its argument, $\omega t$, changes by $2\pi$. So, the period T is given by $\omega T = 2\pi$, or $T = 2\pi/\omega$. The quantity $\omega$ is called the angular frequency.

Simple Harmonic Motion (SHM)

Simple harmonic motion (SHM) is a special type of periodic motion where the displacement of a particle from its equilibrium (mean) position is a sinusoidal function of time.

The displacement $x(t)$ as a function of time $t$ is given by the equation:

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

where the terms are defined as:

• $x(t)$: The displacement from the mean position at time $t$.
• $A$: The amplitude, which is the magnitude of the maximum displacement from the mean position. The motion is confined between $-A$ and $+A$.
• $\omega$: The angular frequency, related to the period $T$ and frequency $\nu$ by $\omega = 2\pi/T = 2\pi\nu$. Its unit is radians per second (rad/s).
• $(\omega t + \phi)$: The phase of the motion at time $t$. It determines the state of the particle (its position and velocity) at that instant.
• $\phi$: The phase constant or initial phase. It is the phase at $t=0$ and determines the initial position of the particle.

Simple Harmonic Motion and Uniform Circular Motion

There is a simple and elegant connection between SHM and uniform circular motion (UCM). The projection of a particle undergoing uniform circular motion onto any diameter of the circle executes simple harmonic motion.

Consider a particle P moving uniformly in a circle of radius $A$ with constant angular speed $\omega$. The circle is called the reference circle, and the particle P is the reference particle.

Let the initial angle of the particle's position vector be $\phi$ at $t=0$. At a time $t$, the angle becomes $(\omega t + \phi)$.

The projection of the particle's position onto the x-axis, let's call it P', has a coordinate $x(t)$ given by:

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

This is precisely the equation for SHM. Similarly, the projection onto the y-axis, $y(t) = A \sin(\omega t + \phi)$, also represents an SHM.

This connection provides a powerful geometrical tool to visualize and derive the properties of SHM.

Velocity and Acceleration in Simple Harmonic Motion

We can find the velocity and acceleration of a particle in SHM either by taking the projection of the velocity and acceleration of the reference particle in UCM, or by differentiating the displacement equation $x(t) = A \cos(\omega t + \phi)$ with respect to time.

Velocity in SHM

Differentiating the displacement $x(t)$ gives the velocity $v(t)$:

$v(t) = \frac{dx}{dt} = \frac{d}{dt}[A \cos(\omega t + \phi)]$

$v(t) = -A\omega \sin(\omega t + \phi)$

Key features of velocity in SHM:

• The velocity is also a sinusoidal function of time, but it is out of phase with the displacement by $\pi/2$ radians (a quarter cycle).
• The maximum speed, or velocity amplitude, is $v_{max} = A\omega$. This occurs when the particle passes through the mean position ($x=0$).
• The speed is zero at the extreme positions ($x = \pm A$).

The velocity can also be expressed as a function of position $x$:

$v(t) = \pm \omega \sqrt{A^2 - x(t)^2}$

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Acceleration in SHM

Differentiating the velocity $v(t)$ gives the acceleration $a(t)$:

$a(t) = \frac{dv}{dt} = \frac{d}{dt}[-A\omega \sin(\omega t + \phi)]$

$a(t) = -A\omega^2 \cos(\omega t + \phi)$

Since $x(t) = A \cos(\omega t + \phi)$, we can write this as:

$a(t) = -\omega^2 x(t)$

Key features of acceleration in SHM:

• The acceleration is directly proportional to the displacement.
• The negative sign indicates that the acceleration is always directed opposite to the displacement, i.e., always towards the mean position.
• The acceleration is maximum in magnitude, $a_{max} = A\omega^2$, at the extreme positions ($x = \pm A$).
• The acceleration is zero at the mean position ($x=0$).

Force Law for Simple Harmonic Motion

According to Newton's second law, $F = ma$. Substituting the expression for acceleration in SHM, $a = -\omega^2 x$, we find the force law for SHM.

$F(t) = m a(t) = m(-\omega^2 x(t))$

$F(t) = -k x(t)$

where $k = m\omega^2$ is a positive constant called the force constant or spring constant.

This provides an alternative and equivalent definition of SHM: Simple harmonic motion is the motion executed by a particle subject to a force that is directly proportional to its displacement from the mean position and is always directed towards the mean position. This type of force is a linear restoring force.

From this force law, we can express the angular frequency and the period of SHM in terms of the system's physical properties (mass $m$ and force constant $k$):

$\omega = \sqrt{\frac{k}{m}} \quad \text{and} \quad T = 2\pi\sqrt{\frac{m}{k}}$

Energy in Simple Harmonic Motion

For a particle executing SHM under the restoring force $F = -kx$, both kinetic and potential energy vary with time and position, but the total mechanical energy remains constant.

Potential Energy (U)

The force $F = -kx$ is a conservative spring-like force. The elastic potential energy stored in the system is:

$U = \frac{1}{2}kx^2 = \frac{1}{2}kA^2 \cos^2(\omega t + \phi)$

Potential energy is zero at the mean position ($x=0$) and maximum at the extreme positions ($x=\pm A$).

Kinetic Energy (K)

The kinetic energy of the oscillating particle is:

$K = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2 A^2 \sin^2(\omega t + \phi)$

Using $k=m\omega^2$, we can also write this as:

$K = \frac{1}{2}kA^2 \sin^2(\omega t + \phi)$

Kinetic energy is maximum at the mean position ($x=0$) and zero at the extreme positions ($x=\pm A$).

Total Mechanical Energy (E)

The total energy is the sum of the kinetic and potential energies:

$E = K + U = \frac{1}{2}kA^2 \sin^2(\omega t + \phi) + \frac{1}{2}kA^2 \cos^2(\omega t + \phi)$

Using the identity $\sin^2\theta + \cos^2\theta = 1$:

$E = \frac{1}{2}kA^2 = \text{Constant}$

The total mechanical energy in SHM is constant and is proportional to the square of the amplitude. During the oscillation, energy is continuously transformed between potential and kinetic forms.

Some Systems Executing Simple Harmonic Motion

1. Oscillations of a Spring-Mass System

A block of mass $m$ attached to a massless spring with spring constant $k$, oscillating on a frictionless horizontal surface, is a classic example of a simple harmonic oscillator. The restoring force is given by Hooke's Law, $F = -kx$.

This perfectly matches the force law for SHM. Therefore, the motion is SHM with an angular frequency and period given by:

$\omega = \sqrt{\frac{k}{m}} \quad \text{and} \quad T = 2\pi\sqrt{\frac{m}{k}}$

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2. The Simple Pendulum

A simple pendulum consists of a point mass (the bob) of mass $m$ suspended from a rigid support by a massless, inextensible string of length $L$.

When the bob is displaced by an angle $\theta$ from the vertical, the force of gravity ($mg$) has a component perpendicular to the string, which acts as the restoring force:

$F_{restoring} = -mg \sin\theta$

This force is not directly proportional to the displacement (which is the arc length $s=L\theta$). Therefore, the motion of a simple pendulum is generally not SHM.

Small Angle Approximation

However, if the angle $\theta$ is small (typically less than about 10 degrees), we can use the approximation $\sin\theta \approx \theta$ (where $\theta$ is in radians). The restoring force then becomes:

$F \approx -mg\theta = -mg\left(\frac{s}{L}\right) = -\left(\frac{mg}{L}\right)s$

Now, the force is proportional to the displacement $s$. This is the condition for SHM. The effective force constant is $k = mg/L$.

The period of a simple pendulum for small oscillations is:

$T = 2\pi\sqrt{\frac{m}{k}} = 2\pi\sqrt{\frac{m}{mg/L}} = 2\pi\sqrt{\frac{L}{g}}$

Notably, for small amplitudes, the period of a simple pendulum is independent of its mass and amplitude.

Damped Simple Harmonic Motion

In real-world systems, oscillations do not continue forever. Frictional or drag forces, collectively known as damping forces, are always present. These forces oppose the motion and cause the mechanical energy of the system to be dissipated, usually as heat. As a result, the amplitude of the oscillations gradually decreases to zero.

The damping force is often proportional to the velocity of the oscillator and acts in the opposite direction:

$F_d = -bv$

where $b$ is the damping constant.

The total force on the oscillator is the sum of the restoring force and the damping force: $F = -kx - bv$. The equation of motion becomes a second-order differential equation:

$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$

For small damping, the solution to this equation shows that the motion is still oscillatory, but the amplitude decreases exponentially with time:

$x(t) = A_0 e^{-bt/2m} \cos(\omega' t + \phi)$

where $A_0$ is the initial amplitude. The amplitude at any time $t$ is $A(t) = A_0 e^{-bt/2m}$. The angular frequency $\omega'$ is slightly less than the natural angular frequency $\omega = \sqrt{k/m}$.

Forced Oscillations and Resonance

When a damped oscillator is subjected to an external periodic driving force, it undergoes forced oscillations. The oscillator eventually stops oscillating at its own natural frequency and is forced to oscillate at the frequency of the external driver, called the driving frequency ($\omega_d$).

The amplitude of these forced oscillations depends on the driving frequency and the amount of damping. The amplitude is given by:

$A = \frac{F_0/m}{\sqrt{(\omega^2 - \omega_d^2)^2 + (b\omega_d/m)^2}}$

where $F_0$ is the amplitude of the driving force and $\omega$ is the natural frequency of the oscillator.

Resonance

The amplitude of the forced oscillations becomes very large when the driving frequency $\omega_d$ is close to the natural frequency $\omega$ of the oscillator. This phenomenon is called resonance.

At resonance ($\omega_d \approx \omega$), the system absorbs the maximum amount of energy from the driving force, leading to a large amplitude of oscillation. The height and sharpness of the resonance peak depend on the amount of damping. Less damping leads to a higher and sharper peak.

Resonance is a very important phenomenon with numerous applications (e.g., tuning a radio, microwave ovens) and potential dangers (e.g., the collapse of bridges due to marching soldiers or wind, earthquake damage to buildings).

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Additional Exercises

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