Periodic Motion

1. Introduction

A motion that repeats itself at equal intervals of time is called periodic motion (for example, the motion of the hands of a clock, the Earth around the Sun, or a vibrating tuning fork). When a periodic motion is a to-and-fro motion about a fixed mean position, it is called oscillatory or vibratory motion. The most important and simplest kind of oscillatory motion is simple harmonic motion (SHM), which is the foundation for understanding pendulums, springs, and waves.

This chapter defines SHM and its equation, derives the energy of a body in SHM, obtains the period of a mass on a spring (horizontal and vertical) and of a simple pendulum, introduces angular SHM, and describes damped and forced oscillations and resonance.

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2. Simple Harmonic Motion

Definition: Simple harmonic motion is the oscillatory motion of a body in which the acceleration is always directed towards a fixed mean position and is directly proportional to the displacement of the body from that mean position.

$$a = -\omega^2 y$$

Here $y$ is the displacement from the mean position and $\omega^2$ is a positive constant. The negative sign shows that the acceleration (and the restoring force) is always directed opposite to the displacement, that is, towards the mean position.

2.1 Equation of SHM (displacement)

SHM can be viewed as the projection, on a diameter, of a particle moving in a circle of radius $r$ (the amplitude) with constant angular velocity $\omega$. If the reference particle turns through angle $\theta = \omega t$, the displacement of its projection is:

$$y = r\sin\omega t$$

where $r$ is the amplitude (maximum displacement) and $\omega$ is the angular frequency. The time period and frequency are:

$$ \begin{aligned} T &= \frac{2\pi}{\omega} \ \qquad f &= \frac{1}{T} \ &= \frac{\omega}{2\pi} \end{aligned} $$

2.2 Velocity in SHM

Differentiating $y = r\sin\omega t$ with respect to time:

$$ \begin{aligned} v &= \frac{dy}{dt} \ &= r\omega\cos\omega t \ &= r\omega\sqrt{1 - \sin^2\omega t} \end{aligned} $$

Since $\sin\omega t = \dfrac{y}{r}$:

$$v = \omega\sqrt{r^2 - y^2}$$

• At the mean position ($y = 0$): $v = \omega r$ (maximum).
• At the extreme position ($y = r$): $v = 0$.

2.3 Acceleration in SHM

Differentiating the velocity:

$$ \begin{aligned} a &= \frac{dv}{dt} \ &= -r\omega^2\sin\omega t \ &= -\omega^2 y \end{aligned} $$

• At the mean position ($y = 0$): $a = 0$.
• At the extreme position ($y = r$): $a = \omega^2 r$ (maximum in magnitude).

This confirms the defining relation $a = -\omega^2 y$.

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3. Energy in Simple Harmonic Motion

Consider a particle of mass $m$ executing SHM of amplitude $r$ and angular frequency $\omega$.

3.1 Kinetic energy

Using $v = \omega\sqrt{r^2 - y^2}$:

$$ \begin{aligned} \text{K.E.} &= \frac{1}{2}mv^2 \ &= \frac{1}{2}m\omega^2\left(r^2 - y^2\right) \end{aligned} $$