Key Points

Periodic motion is any motion that repeats itself at regular time intervals. Oscillatory motion is a specific type of periodic motion that involves moving back and forth about a mean or equilibrium position.

The period (T) is the smallest time interval after which motion is repeated, measured in seconds (s). Frequency (v) is the number of repetitions per unit time, given by $v = \\frac{1}{T}$ and measured in Hertz (Hz), where $1 \\text{ Hz} = 1 \\text{ s}^{-1}$.

SHM is the simplest form of oscillatory motion where the restoring force is directly proportional to the displacement from the mean position and is always directed towards it. This is described by the force law $F = -kx$.

The displacement $x$ of a particle executing SHM at time $t$ is described by the sinusoidal function $x(t) = A \\cos(\\omega t + \\phi)$. Here, $A$ is amplitude, $\\omega$ is angular frequency, and \phi is the phase constant.

Amplitude (A) is the maximum displacement from the mean position. Angular frequency ($\\omega$) is related to the period by $\\omega = \\frac{2\\pi}{T} = 2\\pi v$. The phase ($\\omega t + \\phi$) determines the state of motion at any time $t$.

The velocity of a particle in SHM is given by $v(t) = -\\omega A \\sin(\\omega t + \\phi)$. The speed is maximum ($v_{\\text{max}} = \\omega A$) at the mean position ($x=0$) and zero at the extreme positions.

The acceleration in SHM is given by $a(t) = -\\omega^2 A \\cos(\\omega t + \\phi)$, which simplifies to $a(t) = -\\omega^2 x(t)$. Acceleration is always directed towards the mean position and is maximum ($a_{\\text{max}} = \\omega^2 A$) at the extremes.

Using Newton's second law, the force in SHM is $F = ma = -m\\omega^2 x$. Comparing this with Hooke's law ($F = -kx$), the spring or force constant is $k = m\\omega^2$.

For a mass $m$ attached to a spring with spring constant $k$, the angular frequency is $\\omega = \\sqrt{\\frac{k}{m}}$. The period of oscillation is $T = 2\\pi \\sqrt{\\frac{m}{k}}$.

The potential energy (U) of a particle in SHM is $U = \\frac{1}{2} kx^2 = \\frac{1}{2} kA^2 \\cos^2(\\omega t + \\phi)$. It is maximum at the extreme positions ($x = \\pm A$) and zero at the mean position.

The kinetic energy (K) of a particle in SHM is $K = \\frac{1}{2} mv^2 = \\frac{1}{2} kA^2 \\sin^2(\\omega t + \\phi)$. It is maximum at the mean position ($x=0$) and zero at the extreme positions.

The total mechanical energy $E = K + U$ of a particle in SHM is constant, assuming no damping forces. The total energy is given by $E = \\frac{1}{2} kA^2$.

Simple Harmonic Motion can be viewed as the projection of uniform circular motion onto a diameter of the circle. The particle in circular motion is called the reference particle and the circle is the reference circle.

A simple pendulum consists of a point mass (bob) suspended by a massless, inextensible string. For small angular displacements ($\\theta$), its motion is approximately simple harmonic.

The time period of a simple pendulum of length $L$ is given by $T = 2\\pi \\sqrt{\\frac{L}{g}}$, where $g$ is the acceleration due to gravity. The period is independent of the mass of the bob and the amplitude for small oscillations.