Simple Harmonic Motion

Introduction to SHM

Periodic motion $\\$ Motion of a particle such that its position gets repeated itself after certain interval of time. Examples: Revolution of earth around sun, rotation of earth about its polar axis

Oscillatory or Vibratory Motion: $\\$ Motion of a particle/body to and fro or back and forth respectively about a fixed point (mean position) in definite interval of time

Simple Harmonic Motion(SHM): $\\$ $\to$ The periodic motion, in which a particle moves to and fro repeatedly about a mean position under action of certain restoring force which is directed towards the mean position. $\\$

$\to$ The magnitude of restoring force and acceleration in SHM at any instant is directly proportional to the displacement of particle from the mean position of that instant. $\\$ $\\$ $a\ \alpha\ –y\\$ or, $\ F\ \alpha\ –y\\$ so, $\ F\ =\ -ky\\$ (K) is known as force constant, and –ve sign shows that the restoring force is directed towards the mean position.

Note:

Oscillatory Motions are necessarily periodic motion but all periodic motion are not necessarily oscillatory motions. $\\$ Similarly, all SHM are periodic but all periodic motions may not be simple harmonic motion

Representation of SHM: $\\$ SHM can be represented by the projection /shadow on y-axis such that the body is assumed to be moving in a circular path. The distance between projection of body on y-axis and center of circle represents the displacement of SHM from mean position.

Equations for Linear SHM

Displacement(y): $\\$ $y = r \sin(\omega t) = r \sin(2π/T\times t) \\$ where y is displacement of particle from mean position at any time t , r is amplitude of motion and T is time period

Velocity (v): $\\$ $v = \dfrac{dy}{dt} = r\omega \cos(\omega t) = \omega r \sin(\omega t+π/2) = \omega \sqrt{r^2 –y^2}$

Acceleration (a): $\\$ $a\ =\ \dfrac{dv}{dt} =\ -\omega^2\ r\ \sin(\omega t)\ =\ \omega^2\ r\ \sin(\omega t-\pi)\ =\ -\omega^2\ y$

Time period (T): $\\$ $T = 2\pi\sqrt{\dfrac{y}{a}\ } = 2\pi\ \sqrt{\left(\dfrac{displacement}{acceleration}\right)}$

Frequency (f) $\\$ $f = 1/T = \frac{1}{2\pi}\ \sqrt{\frac{a}{y}\ \ }=\frac{1}{2\pi}\ \sqrt{\frac{acceleration}{displacement}}\$

Angular frequency ( $\omega$ ): $\\$ $\omega=\sqrt{\dfrac{acceleration}{displacement}} = \sqrt{\dfrac{a}{y}} \\$ Also, $\omega = 2πf = 2π/T$

Differential equation for linear SHM: $\\$ $a α –y$ ----------- (i) $\\$ or, $a = -ky$ ------------(ii) $\\$ $\dfrac{d^2y}{dx^2} + ky = 0$ --------------(iii) $\\$ here, equation (iii) is the differential equation for linear SHM

NOTE:

$V_{max} = r\omega$ (at mean position; y=0) $\\$ $V_{min} = 0$ (at extreme position; y=r) $\\$ $a_{max} = -\omega^2r$ ( at extreme position; y=r) $\\$ $a_{min} = 0$ ( at mean position; y=0)

Energy in SHM

Kinetic Energy: $\\$ The kinetic energy in SHM is due to to and fro motion of the particle executing SHM. The expression for kinetic energy is: $\\$ $K.E = \dfrac{1}{2} mv^2 = \dfrac{1}{2} m\omega^2(r^2-y^2)$

Potential Energy: $\\$ The potential energy in SHM is due to restoring force directed towards mean position. The expression for Potential Energy is: $U = \dfrac{1}{2} ky^2 = \dfrac{1}{2} m\omega^2y^2$ where $[ \omega= \sqrt{k/m} ]$

Total Energy: $\\$ The value of total energy in SHM for an oscillating system is always constant. $\\$ $TE = KE + U \\ = \dfrac{1}{2} mw^2(r^2-y^2) + \dfrac{1}{2} mw^2y^2 = \dfrac{1}{2} mw^2r^2 \\$

This equation shows that TE for a system is always constant as $m,\omega$ and $r$ for particular system is always constant

The graph alongside shows Total Energy (straight line) , Potential Energy(dotted curve) and Kinetic Energy(continuous curve). $\\$ It is clear from the graph that:

Total Energy is always constant regardless of the position of particle
Potential Energy graph follows parabolic path, It is maximum at extreme position and minimum (0) at mean position.
Kinetic Energy graph also follows parabolic path, it is maximum at mean position and minimum at extreme position.
Total energy is Potential in nature at extreme position and Kinetic in nature at mean position.

Graphs in SHM

The graph of displacement(y), velocity(v) , acceleration, force, momentum with time all are sine curves.
The velocity-displacement (v-y) graph for simple harmonic motion y= r sinwt and v=r coswt is ellipse.
Displacement(y)-acceleration(a) and displacement(y)-force(F) graph is straight line.

Simple Pendulum

It consists of a heavy point mass (bob) which is suspended by a light, inextensible string fixed from a rigid surface in such a way that the bob is free to oscillate simple harmonically to and fro with respect to a fixed position (mean position).

The motion of simple pendulum is simple harmonic because the acceleration produced on bob is directly proportional to its displacement from mean position and is directed towards it.
Restoring Force $(F) = -mg \sinθ \\$ As θ is very small, $\sinθ ≈ θ \\$ Thus, $F = -mgθ = -mg y/l$