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SIMPLE HARMONIC MOTION

Understanding simple harmonic motion (SHM)

    ➤ Repetitive back and forth movement through an equilibrium position.

    ➤ Maximum displacement on either side of the equilibrium position is equal.

    ➤ The time interval of each complete vibration (to and fro motion) is the same.

Mathematical Representation of Simple Harmonic Motion

Any motion that repeats at regular intervals is called periodic motion or harmonic motion. Here we are interested in a periodic motion of simple harmonic motion (SHM) type which can be represented using a sinusoidal function of time \(t\). In simple harmonic motion (SHM), the displacement \(x(t)\) of a particle from its equilibrium position is described by the equation $$ x = x_{m} \cos(\omega t + \phi) $$ in which \(x_{m}\) is the amplitude of the displacement, \((\omega t + \phi)\) is the phase of the motion, and \(\phi\) is the phase constant. The angular frequency \(\omega\) is related to the period and frequency of the motion by $$\omega = \frac{2\pi}{T} = 2\pi f$$ Differentiating \(x(t)\) leads to equations for the particle's SHM velocity and acceleration as functions of time:

velocity $$v = - \omega x_{m} \sin(\omega t + \phi) $$

acceleration $$ a = -\omega^{2} x_{m} \cos(\omega t + \phi) $$ $$\begin{aligned} a &= -\omega^{2} x_{m} \cos(\omega t + \phi) \\ a &= -\omega^{2} x\\ a &\propto -x \end{aligned}$$

    ➤ In SHM, the acceleration is proportional to the position of the body and its direction is opposite the direction of the displacement from equilibrium position.

In the velocity function, the positive quantity \(\omega x_{m} \) is the velocity amplitude \(v_{m} \). In the acceleration function, the positive quantity \(\omega^{2} x_{m} \) is the acceleration amplitude \(a_{m} \).

SHM in a Spring

    ➤ In SHM, the net force of the system, is proportional to the displacement and acts in the opposite direction of the displacement.

A particle with mass \(m\) that moves under the influence of a Hooke's law restoring force given by \(F = -kx\) is a linear simple harmonic oscillator with

Angular frequency, $$\omega = \sqrt \frac{k}{m} $$

and Time Period, $$ T = \frac{2\pi}{\omega} = 2\pi\sqrt \frac{m}{k} $$

SHM in a Simple Gravity Pendulum

    ➤ Here, the net weight of the bob acts downwards and the tension acts along the pendulum string. The tension 'T' depends on the amplitude but the period of oscillation is independent of the mass of the bob and the amplitude of the swing.

Period of oscillation of a simple pendulum depends only on the length of the string. \[ T = 2 \pi \sqrt \frac L g\]