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:

1. Total Energy is always constant regardless of the position of particle
2. Potential Energy graph follows parabolic path, It is maximum at extreme position and minimum (0) at mean position.
3. Kinetic Energy graph also follows parabolic path, it is maximum at mean position and minimum at extreme position.
4. 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,$F = -mgθ = -mg y/l$

• Acceleration: $a=F/m = -(g/l).y \\$

• Angular frequency($\omega$) : $\omega =\sqrt{\dfrac{acceleration}{displacement}} = \sqrt{\dfrac{a}{y}}\ =\sqrt{\dfrac{g}{l}}$

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Important Points Regarding Time Period of Simple Pendulum

$\\$

$T = 2π\sqrt{\dfrac{l}{g}}$ where, $l$= length of pendulum, $g$= acceleration due to gravity$\\$$T α \sqrt{l}$: (law of length)$\\ T α \dfrac{1}{\sqrt{g}}$: (law of gravity)$\\$$\\$

• Time period of pendulum is independent of amplitude (law of isochronism).
• Time period of pendulum is independent of mass of bob.
• For oscillating pendulum, $T-L$ graph is parabolic and $T^2-L$ graph is straight line.
• The pendulum of time period 2 second is called second pendulum. To obtain a second pendulum in place of $g = 9.8 m/s^2$ , we need pendulum of length almost equal to 1m .

Time Period of Simple Pendulum

Pendulum in a lift

• When a lift with oscillating simple pendulum moves at constant velocity upward, then$\\$$T = 2π\sqrt{\dfrac{l}{g}} \\$
• When lift with oscillating simple pendulum moves upwards at constant acceleration (a), then$\\$$T = 2π\sqrt{\dfrac{l}{g+a}} \\$
• When lift accelerates uniformly downward with acceleration(a), then$\\$$T = 2π\sqrt{\dfrac{l}{g-a}}$ ( a< g)$\\$

$\\$

NOTE:

If a = g, then T =∞ (pendulum doesn't oscillate)$\\$If a>g , then center of oscillation will be above point of suspension$\\$$T = 2π\sqrt{\dfrac{l}{a-g}}$

Pendulum of long string

When length of simple pendulum $'l'$ is comparable to radius of earth $'R'$, then$\\$$T=2\pi \sqrt{\dfrac{R}{g\left(1+\frac{R}{L}\right)}} \\$

• If $L = R$, then $T=2 \pi \sqrt{ \dfrac{R}{2g } }$ = 59.8 minutes ≈ 1 hour
• If Length is very large, i.e. $L → \infty$, then $T=2\pi \sqrt{\dfrac{R}{g}}\$ = 84.6 minutes

Motion of Pendulum inside fluid

When a simple pendulum with bob (density = ρ) oscillates in non-viscous fluid of density (σ) , then$\\$$g_{eff} =\left(1-\dfrac{\sigma}{\rho}\right)g$ ; where: ( ρ > σ )$\\$so,$T = 2π\sqrt{\dfrac{l}{\left(1-\frac{\sigma}{\rho}\right)g\ \ }} \\$
$\\$Note:$\\$

if ρ < σ ; the bob will float up the fluid and oscillation cannot be carried out

In a vehicle accelerating along horizontal road:

If acceleration due to gravity at that place is 'g' and 'a' be the acceleration of the vehicle, then$\\$$a_{eff} = \sqrt{a^2+g^2\ } \\$so,$T=2\pi\ \sqrt{\dfrac{l}{a_{eff}}}$

Spring Mass

Horizontal Spring System$\\ \\$

In this system, the mass attached oscillates horizontally.$\\$The time period of oscillation of horizontal pendulum is: $T=2\pi\sqrt{\dfrac{m}{k}\ }\ \\$Where, m being mass of body attached and K is spring constant

Vertical Spring System$\\ \\$

In this system, the mass attached oscillates horizontally$\\$And, time period of oscillation is same as horizontal i.e. $T=2\pi\sqrt{\dfrac{m}{k}\ }\ \\$where the symbols have their usual meaning

Combination of Springs

Series Combination of Springs

Force applied in each spring remains equal i.e. $F_1 = F_2 = F\\$Extension will be different so, Total extension: $y= y_1 + y_2 \\$

Effective spring constant of system in series combination (keff) is related as:$\\$$\dfrac{1}{k_{eff}}=\dfrac{1}{k_1}+\dfrac{1}{k_2}$

Parallel Combination of Springs

Force on individual springs will be different Extensions will be equal :$y_1 = y_2 = y\\$Net force is : $F = F_1 + F_2\\$

Effective spring constant of the paralle combination (keff) is related as:$\\ K_{eff} = K_1 + K_2$

Points related to Spring system:

• If the mass of spring is also considered in vertical spring sytem:$\\$$T=2\pi\ \sqrt{\dfrac{m+\frac{m_s}{3}}{k\ }\ } \\$

• If two masses m1 and m2 are connected at either end of spring then.$\\$Time period $(T)=2\pi\sqrt{\dfrac{m_{eff}}{k}\ } \\$

Where, $m_{eff} = \dfrac{m_1m_2}{m_1+m_2}$ known as reduced mass.

• When a spring of force constant k is divided into n equal parts, then force constant of each part becomes nk,

And, the force constant of their parallel combination becomes $n^2k$.

Other Types of Pendulum

Compound Pendulum

Any rigid body if suspended is capable to swing in vertical plane about axis passing through it, this system of oscillating rigid body forms a compound pendulum.$\\$Time period for compound pendulum is : $T=2\pi\sqrt{\dfrac{I}{mgd}\ } \\$Where, I = Moment of inertia of body, d = Distance of C.G from point of suspension

Torsional Pendulum

Rigid body is capable to show angular simple harmonic motion about axis passing through body, then it is called torsional pendulum.$\\$Time period for torsional pendulum is : $T=2\pi\sqrt{\dfrac{I}{C}\ } \\$; where C is Torsional Rigidity

Better You Know

• Time period of body oscillating from a spring is independent of acceleration due to gravity (g).
• When acceleration of a particle executing SHM increases, then its time period remains same.
• Distance and Work done by simple pendulum in one complete oscillation is zero, while distance = 4r (r being amplitude).
• The time period of simple pendulum increases during summer i.e. it looses time
• The time period of simple pendulum decreases during winter i.e. it gains time
• The cause of change of time period of pendulum during seasonal change is due to thermal expansion of string.
• Change in amplitude of oscillation of simple pendulum doesn't changes its time period.
• A particle executing SHM with a frequency of f has its KE oscillating with frequency of 2f.