physics_oscillation_progressive_waves
 Oscillation and Progressive Waves Periodic Motion: Periodic Motion is motion which repeats itself in equal interval of time. Oscillations or vibrations1. If a body undergoing periodic motion has an equilibrium position somewhere inside its path where no net external force acts on the body, then such motion is called oscillations or vibrations.2. the body is left on equilibrium position left at rest, then it remains there forever. If the body is given a small displacement from the equilibrium position, a force comes into play which tries tobring the body back to the equilibrium point. Such motion is called oscillations or vibrations. 1. Every oscillatory motion is periodic, but every periodic motion need not be oscillatory.2. Circular motion is a periodic motion, but it is not oscillatory. Simple Harmonic motionSimple harmonic motion (SHM) is a periodic motion in which displacement is a sinusoidal function of time. Simple Harmonic motionSimple Harmonic motion is defined as linear period motion of a body, in which the restoring force (or acceleration) is always directed towards the mean position and its magnitude is directly proportional to the displacement from the mean position. a d2xdt2+𝜔2x=0⟹vdvdx=-𝜔2x⟹v22=-𝜔2x22+C At x=± a , v=0⟹ C=𝜔2a22⟹v2=𝜔2(a2-x2) $\begin{array}{l}\phantom{\rule{1.76em}{0ex}}\frac{{d}^{2}x}{d{t}^{2}}+{𝜔}^{2}x=0\\ ⟹v\frac{dv}{dx}=-{𝜔}^{2}x\\ ⟹\frac{{v}^{2}}{2}=-\frac{{𝜔}^{2}{x}^{2}}{2}+C\\ \phantom{\rule{1.76em}{0ex}}At\phantom{\rule{0.22em}{0ex}}x=±\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}},\phantom{\rule{0.22em}{0ex}}v=0\\ ⟹\phantom{\rule{0.44em}{0ex}}C=\frac{{𝜔}^{2}{a}^{2}}{2}\\ ⟹{v}^{2}={𝜔}^{2}\left({a}^{2}-{x}^{2}\right)\\ \end{array}$ Amplitude PeriodThe smallest interval of time after which the motion is repeated is called its period. Frequency Wavelength Velocity Composition of SHM having same time Periodx1=A1sin(𝜔t+𝜙1)${x}_{1}={A}_{1}sin\left(𝜔t+{𝜙}_{1}\right)$x2=A2sin(𝜔t+𝜙2)${x}_{2}={A}_{2}sin\left(𝜔t+{𝜙}_{2}\right)$ x=x1+x2$x={x}_{1}+{x}_{2}$x=R sin(𝜔t+𝛿)$x=R\phantom{\rule{0.22em}{0ex}}sin\left(𝜔t+𝛿\right)$whereR2=A21+A22+2A1A2 cos(𝜙1-𝜙2)${R}^{2}={A}_{1}^{2}+{A}_{2}^{2}+2{A}_{1}{A}_{2}\phantom{\rule{0.22em}{0ex}}cos\left({𝜙}_{1}-{𝜙}_{2}\right)$tan𝛿=A1sin𝜙1+A2sin𝜙2A1cos𝜙1+A2cos𝜙2$tan𝛿=\frac{{A}_{1}sin{𝜙}_{1}+{A}_{2}sin{𝜙}_{2}}{{A}_{1}cos{𝜙}_{1}+{A}_{2}cos{𝜙}_{2}}$ Time Period of Simple PendulumT=2𝜋 L g $T=2𝜋\sqrt{\phantom{\rule{0.22em}{0ex}}\frac{L\phantom{\rule{0.22em}{0ex}}}{g\phantom{\rule{0.22em}{0ex}}}}$ Second’s PendulumA simple pendulum whose period is two seconds is called second’s pendulum. Energy of a Particle Performing S.H.MKinetic EnergyEk=12m𝜔2(A2-x2)=12k(A2-x2) ${E}_{k}=\frac{1}{2}m{𝜔}^{2}\left({A}^{2}-{x}^{2}\right)=\frac{1}{2}k\left({A}^{2}-{x}^{2}\right)\phantom{\rule{0.22em}{0ex}}$ Potential EnergyEp=12m𝜔2x2=12kx2 ${E}_{p}=\frac{1}{2}m{𝜔}^{2}{x}^{2}=\frac{1}{2}k{x}^{2}\phantom{\rule{0.22em}{0ex}}$ Damped Simple Harmonic Motionmd2xdt2+bdxdt+c=0 $m\frac{{d}^{2}x}{d{t}^{2}}+b\frac{dx}{dt}+c=0\phantom{\rule{0.22em}{0ex}}$wherem = mass of objectk = spring constantb = depends on characteristics of the medium (viscosity). x(t)=Ae- bt2mcos(𝜔't+𝜙)$x\left(t\right)=A{e}^{-\phantom{\rule{0.22em}{0ex}}\frac{bt}{2m}}cos\left(𝜔\text{'}t+𝜙\right)$𝜔'=km-b24m2 $𝜔\text{'}=\sqrt{\frac{k}{m}-\frac{{b}^{2}}{4{m}^{2}}\phantom{\rule{0.22em}{0ex}}}$ Mechanical energy of the undamped oscillatorE(t)=12kA2e- bt m $E\left(t\right)=\frac{1}{2}k{A}^{2}{e}^{-\phantom{\rule{0.22em}{0ex}}\frac{bt}{\phantom{\rule{0.22em}{0ex}}m\phantom{\rule{0.22em}{0ex}}}}$ Forced Oscillation and Resonancemd2xdt2+bdxdt+c=F0cos𝜔dt $m\frac{{d}^{2}x}{d{t}^{2}}+b\frac{dx}{dt}+c={F}_{0}cos{𝜔}_{d}t\phantom{\rule{0.22em}{0ex}}$ Periodic force of (angular) frequency 𝜔d${𝜔}_{d}$ is applied. The oscillator, initially, oscillates with its natural frequency𝜔$𝜔$. When we apply the external periodic force, the oscillations with the natural frequency die out, and then the body oscillates with the (angular) frequency of the external periodic force. x(t)=A cos(𝜔dt+𝜙) $x\left(t\right)=A\phantom{\rule{0.22em}{0ex}}cos\left({𝜔}_{d}t+𝜙\right)\phantom{\rule{0.22em}{0ex}}$ A=F0{m2(𝜔2-𝜔2d)+𝜔2d b2}12 $A=\frac{{F}_{0}}{\left\{{m}^{2}\left({𝜔}^{2}-{𝜔}_{d}^{2}\right)+{𝜔}_{d}^{2}\phantom{\rule{0.22em}{0ex}}{b}^{2}{\right\}}^{\frac{1}{2}}}\phantom{\rule{0.22em}{0ex}}$ Progressive wavey=a sin (kx-𝜔t)$y=a\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}\left(kx-𝜔t\right)$ Reflection of WavesReflection is the phenomenon in which progressive wave traveling from one medium to another comes back in the original medium with slightly different intensity and energy. 1. A travelling wave or pulse suffers a phase change of π $\phantom{\rule{0.22em}{0ex}}\pi \phantom{\rule{0.22em}{0ex}}$on reflection at a rigid boundary.2. A travelling wave or pulse suffers no phase change on reflection at an open boundary. Speed of a Transverse Wave on Stretched Stringv=T𝜇$v=\sqrt{\frac{T}{𝜇}}$ Speed of longitudinal waves in a mediumv=B𝜌$v=\sqrt{\frac{B}{𝜌}}$ Speed of longitudinal waves in a Linear solid barv=Y𝜌$v=\sqrt{\frac{Y}{𝜌}}$For a linear medium, like a solid bar, the lateral expansion of the bar is negligible. So we consider it to be only under longitudinal strain. So, the relevant modulus of elasticity is Young’s modulus Standing wavesy1=a sin (kx-𝜔t)${y}_{1}=a\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}\left(kx-𝜔t\right)$y2=a sin (kx+𝜔t)${y}_{2}=a\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}\left(kx+𝜔t\right)$ y=y1+y2$y={y}_{1}+{y}_{2}$y=a sin (kx-𝜔t)+a sin (kx+𝜔t)$y=a\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}\left(kx-𝜔t\right)+a\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}\left(kx+𝜔t\right)$y=2a sinkx cos𝜔t$y=2a\phantom{\rule{0.22em}{0ex}}sinkx\phantom{\rule{0.22em}{0ex}}cos𝜔t$ Nodes : The points at which the amplitude is zero (i.e., where there is no motion at all) are nodesAntinodes : The points at which the amplitude is the largest are called antinodes. Pitch is considered as frequency. BeatsThe alternate waxing and waning of sound after definite intervals of time, due to superposition of two waves of nearly equal frequencies, is called beats y1=a sin 2𝜋n1t${y}_{1}=a\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}2𝜋{n}_{1}t$y2=a sin 2𝜋n2t${y}_{2}=a\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}2𝜋{n}_{2}t$ y=y1+y2$y={y}_{1}+{y}_{2}$ y=2a sin[2𝜋(n1+n22)t] cos[2𝜋(n1-n22)t]$y=2a\phantom{\rule{0.22em}{0ex}}sin\left[2𝜋\left(\frac{{n}_{1}+{n}_{2}}{2}\right)t\right]\phantom{\rule{0.22em}{0ex}}cos\left[2𝜋\left(\frac{{n}_{1}-{n}_{2}}{2}\right)t\right]$ Letn1+n22=n$\frac{{n}_{1}+{n}_{2}}{2}=n$ y=2a cos[2𝜋(n1-n22)t] . sin[2𝜋nt] $y=2a\phantom{\rule{0.22em}{0ex}}cos\left[2𝜋\left(\frac{{n}_{1}-{n}_{2}}{2}\right)t\right]\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}sin\left[2𝜋nt\right]\phantom{\rule{0.22em}{0ex}}$ Frequency of BeatN=n1-n2$N={n}_{1}-{n}_{2}$ Doppler effect in sound
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