Introduction

This chapter explores oscillatory motion, a type of repetitive motion where an object moves back and forth around an equilibrium position. This motion is fundamental to understanding many physical phenomena, including the production of sound by musical instruments, the vibrations of molecules in solids, and the propagation of waves. The chapter introduces key concepts like period, frequency, displacement, amplitude, and phase, which are essential for describing periodic and oscillatory motions.

Periodic And Oscillatory Motions

Periodic motion is defined as motion that repeats itself identically after regular intervals of time. Examples include the motion of planets or the up-and-down movement of a swimmer.

Oscillatory motion is a specific type of periodic motion where an object moves to and fro about a central equilibrium position. A restoring force, which acts to bring the object back to equilibrium, is characteristic of this motion. Simple harmonic motion (SHM) is the simplest form of oscillatory motion, occurring when the restoring force is directly proportional to the displacement from equilibrium and directed towards it.

Period and frequency

The period (T) is the smallest time interval after which a periodic motion repeats itself. Its SI unit is seconds (s).

The frequency (ν) is the number of repetitions (oscillations) per unit time. It is the reciprocal of the period:

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

The SI unit of frequency is hertz (Hz), where 1 Hz = 1 s^-1.

Displacement

Displacement in oscillatory motion refers to the change in a physical property with time. It is often measured as the distance from the equilibrium position. For a particle undergoing SHM, the displacement x(t) can be described by a sinusoidal function of time, such as:

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

where:

• Amplitude (A): The maximum magnitude of displacement from the equilibrium position.
• Angular frequency (ω): Related to the period and frequency by $\omega = 2\pi\nu = 2\pi/T$.
• Phase (ωt + φ): The argument of the cosine function, which determines the state of motion at time t.
• Phase constant (φ): The phase at t = 0, determined by initial conditions.

Any periodic motion can be represented as a superposition of sinusoidal functions of different periods (Fourier analysis).

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a specific type of periodic motion where the displacement from the equilibrium position varies sinusoidally with time. This motion is characterized by a restoring force that is directly proportional to the displacement and always directed towards the equilibrium position.

Simple Harmonic Motion And Uniform Circular Motion

SHM can be visualized as the projection of uniform circular motion onto a diameter of the circle. If a particle P moves uniformly in a circle of radius A with angular speed ω, its projection P' on a diameter executes SHM. The displacement of P' is given by $x(t) = A \cos(\omega t + \phi)$, where A is the radius of the circle (amplitude), ω is the angular speed, and φ is the initial phase angle.

Velocity And Acceleration In Simple Harmonic Motion

The velocity and acceleration of a particle in SHM can be derived from its displacement function $x(t) = A \cos(\omega t + \phi)$:

• Velocity (v): Differentiating displacement with respect to time gives velocity.

$$ v(t) = \frac{dx}{dt} = -A\omega \sin(\omega t + \phi) $$

The velocity amplitude is $v_{max} = A\omega$. Velocity is maximum when displacement is zero and zero at maximum displacement.

• Acceleration (a): Differentiating velocity with respect to time gives acceleration.

$$ a(t) = \frac{dv}{dt} = -A\omega^2 \cos(\omega t + \phi) $$

Substituting $x(t)$, we get:

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

This shows that acceleration is directly proportional to displacement and is always directed towards the equilibrium position. The acceleration amplitude is $a_{max} = A\omega^2$.

Plots of displacement, velocity, and acceleration against time show that they have the same period but differ in phase. Velocity leads displacement by π/2, and acceleration leads displacement by π (or lags by π).

Force Law For Simple Harmonic Motion

According to Newton's second law, the force (F) causing SHM is given by:

$$ F = ma $$

Substituting the expression for acceleration $a = -\omega^2 x$:

$$ F = -m\omega^2 x $$

This force is proportional to the displacement (x) and is always directed towards the equilibrium position (restoring force). We can define a force constant $k = m\omega^2$, so the force law becomes:

$$ F = -kx $$

This linear relationship between force and displacement characterizes SHM and is also known as Hooke's Law for springs. Systems obeying this force law are called linear harmonic oscillators.

Energy In Simple Harmonic Motion

In SHM, energy oscillates between kinetic energy (K) and potential energy (U). For a particle of mass m with velocity v and displacement x, under a conservative force F = -kx:

• Kinetic Energy (K):

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

Kinetic energy is maximum at the equilibrium position (x=0) and zero at the extreme positions (x=±A).

• Potential Energy (U):

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

Potential energy is zero at the equilibrium position and maximum at the extreme positions.

• Total Energy (E):

The total mechanical energy E = K + U is constant for SHM (in the absence of damping):

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

This means energy is conserved. It continuously transforms between kinetic and potential forms, but the total energy remains constant throughout the motion.

The Simple Pendulum

A simple pendulum consists of a small bob of mass 'm' suspended by a light, inextensible string of length 'L' from a rigid support. For small angular displacements ($\theta$) from the vertical equilibrium position, the motion of the pendulum is approximately simple harmonic.

The restoring torque about the support is $\tau = -L(mg\sin\theta)$. Using Newton's second law for rotation, $\tau = I\alpha$, where $I = mL^2$ is the moment of inertia of the bob:

$$ mL^2 \alpha = -Lmg\sin\theta $$

For small angles ($\sin\theta \approx \theta$), this simplifies to:

$$ \alpha = -\frac{g}{L}\theta $$

This is the equation for SHM in terms of angular displacement. Comparing with $a = -\omega^2 x$, we find the angular frequency of a simple pendulum:

$$ \omega = \sqrt{\frac{g}{L}} $$

The period of oscillation of a simple pendulum for small amplitudes is:

$$ T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{L}{g}} $$

This formula shows that the period of a simple pendulum depends on its length and the acceleration due to gravity, but not on its mass or the amplitude (for small amplitudes).

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