\nIn this chapter, we study oscillatory motion. The description of an oscillatory motion requires some fundamental concept like period, frequency, displacement, amplitude and phase.<\/p>\n
Periodic And Oscillatory Motions<\/span> Oscillations or Vibrations: Note: Frequency (n): Displacement: Explanation: In the case if oscillation of simple pendulum, the angle from the vertical as a function of time may be regarded as a displacement variable. The voltage across a capacitor, changing with time in an a.c. circuit is also a displacement variable<\/p>\n Note: Mathematical expression for displacement: Simple Harmonic Motion (S.H.M.)<\/span> The oscillatory motion is said to be simple harmonic motion if the displacement \u2018x\u2019 of the particle from the origin varies with time as Graphical Variation of S.H.M. The above graph shows the graphical representation of x(t) = A cos wt with time. Amplitude of S.H.M. 2. Simple Harmonic Motion And Uniform Circular Motion<\/span><\/p>\n Question 1. After time Y second, let the particle reach P so that \u2220POP0<\/sub> = \u03c9t. N is the foot of the perpendicular drawn from P on the diameter CD.<\/p>\n Similarly, M is the foot of the perpendicular drawn from P to the diameter AB. When the particle moves along the circumference of the circle, the foot of the perpendicular executes to and fro motion along the diameter CD or AB with O as the mean position. From the right angle triangle O MP, we get Velocity And Acceleration In Simple Harmonic Motion B Case – 1 Acceleration of S.H.M Variation of displacement Y with time t: Variation of acceleration with time: Force Law For Simple Harmonic Motion<\/span> Statement: Energy In Simple Harmonic Motion<\/span> Total energy of the S.H.M. is the sum of the kinetic energy and potential energy. Total energy remains a constant throughout its motion.<\/p>\n Expression for Kinetic energy: Expression for potential energy: Work done to displace the particle through a small distance dy, dw = force \u00d7 displacement Graphical variation of PE, KE and TE of S.H.M. Some Systems Executing Simple Harmonic Motion<\/span> Period of oscillation of a spring: If the body be displaced towards right through a small distance \u2018x\u2019, a restoring force will be developed. The Simple Pendulum: The forces acting on the bob are (1) weight of bob Fg<\/sub> (mg) acting vertically downward. (2) Tension T in the string.<\/p>\n The gravitational force Fg<\/sub> can be divided into a radial component Fg<\/sub>Cos \u03b8 and tangential component Fg<\/sub>Sin \u03b8. The radial component is cancelled by the – tension T. But the tangential component Fg<\/sub>Sin \u03b8 produces a restoring torque.<\/p>\n Restoring torque \u03c4 = – Fg<\/sub> sin \u03b8 . L. Question 2. Let the equilibrium position of block be \u2018O\u2019. If this block is moved along downward direction through a distance x, a damping force will be developed on vane due to liquid. This damping force is proportional to velocity of vane ie; damping force Fd<\/sub> \u03b1 – v (or) Fd<\/sub> = -bv<\/p>\n where b is called damping constant. The value of b depends on the characteristics of the liquid and the vane.<\/p>\n The restoring force on the block due to spring. Fs<\/sub> = -kx. where x is the displacement of the mass from its equilibrium position. The motion of damped harmonic oscillator: Case – 1 Case – 2 Case – 3 The Energy variation of damped oscillator: The above equation shows that the energy of a damped oscillator decreases exponentially with time, which is shown below Note: Forced oscillations and resonance<\/span> Forced oscillation: The differential equation of forced oscillator: The total force acting on the damped oscillator, The motion of forced oscillator: Case -1 Case – 2 Case – 3 This equation shows that the maximum amplitude fora given driving frequency is governed by the driving frequency and damping constant.<\/p>\n Resonant Oscillation<\/span> The amplitude will be maximum when the frequency of the applied periodic force is equal to the natural frequency of the vibration. Such oscillations are called resonant oscillations and the phenomenon is called resonance.<\/p>\n Graphical variation amplitude with driving frequency: Examples of resonance: The Tacoma Narrows Bridge at Puget Sound, Washington, USA was opened on July 1, 1940. Four months later winds produced a pulsating resultant force in resonance with the natural frequency of the structure.<\/p>\n This caused a steady increase in the amplitude of oscillations until the bridge collapsed. It is for the same reason the marching soldiers break steps while crossing a bridge. Aircraft designers make sure that none of the natural frequencies at which a wing can oscillate match the frequency of the engines in flight! Earthquakes cause vast devastation.<\/p>\n In an earthquake, short and tall structures remain unaffected while the medium height structures fall down. This happens because the natural frequencies of the short structures happen to be higher and those of taller structures lower than the frequency of the seismic waves.<\/p>\n Kerala Plus One Physics Notes Chapter 14 Oscillations Introduction In this chapter, we study oscillatory motion. The description of an oscillatory motion requires some fundamental concept like period, frequency, displacement, amplitude and phase. Periodic And Oscillatory Motions Periodic Motion: A motion that repeats itself at regular intervals of time is called periodic motion. Example: The […]<\/p>\n","protected":false},"author":5,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_genesis_hide_title":false,"_genesis_hide_breadcrumbs":false,"_genesis_hide_singular_image":false,"_genesis_hide_footer_widgets":false,"_genesis_custom_body_class":"","_genesis_custom_post_class":"","_genesis_layout":"","footnotes":""},"categories":[42728],"tags":[],"yoast_head":"\n
\nPeriodic Motion:
\nA motion that repeats itself at regular intervals of time is called periodic motion.
\nExample:<\/p>\n\n
\nA body executes to and fro motion at regular intervals of time is called oscillatory (or) vibratory motion.<\/p>\n
\n(1) When the frequency is small, we call it oscillation. When the frequency is high, we call it vibration.
\nPeriod and frequency:
\nPeriod (T):
\nTime taken to complete one oscillation is called period<\/p>\n
\nThe number of oscillations per second is called frequency.
\nfrequency v = \\(\\frac{1}{T}\\)Hz<\/p>\n
\nDisplacement of oscillation means the change of any physical property with time under consideration.<\/p>\n
\nFor example, consider the oscillation of block attached to a spring. In this case the displacement of block with time is referred to as displacement.<\/p>\n
\nThe displacement variable may take both positive and negative values.<\/p>\n
\nThe displacement of a periodic function can be written as
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-1.png\")
<\/p>\n
\nSimple harmonic motion is the simplest form of oscillatory motion.<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-2.png\")
\nWhere
\nx(t) = displacement x as a function of time t
\nA = amplitude
\n\u03c9 = angular frequency
\n(\u03c9 t+ \u03a6) = phase (time-dependent)
\n\u03a6 = phase constant or initial phase<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-3.png\")
<\/p>\n
\n(Initial phase \u03a6 = 0)<\/p>\n
\nThe maximum displacement of S.H.M. from mean position is called the amplitude of S.H.M.
\nNote:
\n1.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-4.png\")
\nTwo simple harmonic motions having, same w and \u03a6 but different amplitudes A and B as shown in the above figure.<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-5.png\")
\nTwo simple harmonic motions having the same A and w but different phase angle \u03a6 as shown in the above figure.<\/p>\n
\nShow that the projection of uniform circular motion on any diameter of the circle is S.H.M.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-6.png\")
\nAnswer:
\nConsider a particle moving along the circumference of a circle of radius \u2018a\u2019 and centre O, with uniform angular velocity w. AB and CD are two mutually perpendicular diameters along X and Y axis. At time t = 0. let the particle be at P0<\/sub> so that \u2220P0<\/sub>OB = \u03a6.<\/p>\n
\ncos (\u03c9t + \u03a6) = \\(\\frac{\\mathrm{OM}}{\\mathrm{OP}}\\)
\n\u2234 OM = OP cos (\u03c9t + \u03a6)
\nX = a cos (\u03c9t + \u03a6) ………………. (1)
\nSimilarly, we get
\nsin (\u03c9t + \u03a6) = \\(\\frac{y}{a}\\) (or)
\nY = a sin (\u03c9t + \u03a6) ……………… (2)
\nEquation (1) and (2) are similar to equations of S.H.M. The equation(1) and (2) shows that the projection of uniform circular motion on any diameter is S.H.M.
\nAt t = 0, if the particle is at B, then \u03a6 = 0. Then equations (1) and (2) reduce to
\nx = a cos \u03c9t ………………….. (3)
\ny = a sin \u03c9t …………………….(4)<\/p>\n
\nVelocity Of S.H.M.
\nThe y displacement of S.H.M. is given by
\ny = a sin \u03c9 t
\n\u2234 velocity in y-direction
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-7.png\")
<\/p>\n
\nAt the mean position y=0, therefore velocity is maximum. The maximum velocity is given by
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-8.png\")
\nCase – 2
\nAt the extreme position, y = a
\n\u2234 Vminimum<\/sub> = \\(\\sqrt{a^{2}-a^{2}}\\) = 0
\nSo the velocity of a S.H.M. varies between o and wa<\/p>\n
\nWe know y = a sin \u03c9 t
\nVelocity v = \\(\\frac{d y}{d t}\\) = a \u03c9 cos \u03c9 t
\nAcceleration a = \\(\\frac{d^{2} y}{d t^{2}}\\) = -a\u03c9\u00b2 sin \u03c9t
\na = -a\u03c9\u00b2 sin \u03c9t
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-9.png\")
\nThis equation shows that the acceleration of an SHM is directly proportional to the displacement and opposite to the displacement.<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-10.png\")
\nVariation of velocity (v) with time:
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-11.png\")
<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-12.png\")
<\/p>\n
\nAccording to Newton’s second law of motion F = ma
\nBut a = -\u03c9\u00b2y(t)
\nie. force acting on the S.H.M. in Y direction
\nF = -m\u03c9\u00b2y(t)
\n(or) F = -ky(t)
\nWhere k= m\u03c9\u00b2
\nFrom the above equation (1), we can take an alternative definition of simple harmonic motion.<\/p>\n
\nSimple harmonic motion is the motion executed by a particle subject to a force, which is proportional to the displacement of the particle and is directed towards the mean position.<\/p>\n
\nA simple harmonically moving particle possesses both potential energy and kinetic energy. Potential energy is due to its displacement against restoring force. Kinetic energy is due to its motion.<\/p>\n
\nLet m be the mass of the particle executing SHM. Let V be the velocity at any instant,
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-13.png\")
<\/p>\n
\nPotential energy is work required to take the particle against the restoring force.<\/p>\n
\n= m\u03c9\u00b2y \u00d7 dy [force = \u03c9\u00b2y ]. Therefore total work done to take the particle from o to y.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-14.png\")
<\/p>\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-15.png\")
\nThis work done is stored in the particle as its potential energy.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-16.png\")
\nAt extreme position y = a
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-17.png\")
\nAt equilibrium position y = 0
\n\u2234 PE = 0
\nTotal energy of an S.H.M.
\nTotal energy = PE + kE
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-18.png\")
\n\u2234 Total energy = maximum KE = maximum PE<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-19.png\")
<\/p>\n
\nOscillations due to a spring Hooks Law:
\nThe force acting simple harmonic motion is proportional to the displacement and is always directed towards the centre of motion.
\nF \u03b1 – x (or) F= kx
\nwhere k is called spring constant<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-20.png\")
\nConsider a body of mass m attached to a massless spring of spring constant K. The other end of spring is connected to a rigid support as shown in figure. The body is placed on a frictionless horizontal surface.<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-21.png\")
<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-22.png\")
\nConsider a mass m suspended from one end of a string of length L fixed at the other end as shown in figure. Suppose P is the instantaneous position of the pendulum. At this instant its string makes an angle \u03b8 with the vertical.<\/p>\n
\n\u03c4 = -mg sin \u03b8.L …………….. (1)
\n-ve sign shown that the torque and angular displacement \u03b8 are oppositely directed. For rotational motion of bob,.
\n\u03c4 = I\u03b1 …………. (2).
\nWhere I is a moment of inertia about the point of suspension and \u03b1 is angular acceleration. From eq (1) and eq (2).
\nI\u03b1 = -mg sin \u03b8.L
\nIf we assume that the displacement \u03b8 is small, sin \u03b8 \u2248 \u03b8.
\n\u2234 I\u03b1 = -mg \u03b8.L
\nI\u03b1 + mg \u03b8.L = 0
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-23.png\")
\nDamped Simple Harmonic Motion<\/span>
\nPeriodic oscillations of decreasing amplitude due to the presence of resistive forces of the medium are called damped oscillations.<\/p>\n
\nDerive a differential equation for a damped simple harmonic oscillation.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-24.png\")
\nAnswer:
\nConsider a block of mass \u2018m\u2019 connected to one end of a massless spring of spring constant K. The other end of spring is connected to rigid support. The block is connected to a vane through a rod (The vane and rod are massless). The vane is submerged in a liquid.<\/p>\n
\n\u2234 Total force on the block, F = -bv + -kx
\nma = -bv + -kx
\nma + bv + kx = 0
\n\\(m \\frac{d^{2} x}{d t^{2}}+b \\frac{d x}{d t}+k x=0\\)
\nThis is the differential equation of S.H.M.<\/p>\n
\nThe solution of the above differential equation of damped harmonic oscillator is
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-25.png\")
<\/p>\n
\nb = 0 (There is no damping force). In this case, we get
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-26.png\")
\nThe above result shows that, if there is no damping force (b = 0). The oscillator behaves like a undamped oscillator.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-27.png\")
<\/p>\n
\nIf b is small, the amplitude of the oscillator decreases continuously with time. The motion is approximately a periodic. The Variation of displacement x (t) with time T is shown below.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-28.png\")
<\/p>\n
\nIf damping constant b is large, the amplitude of the oscillator decreases to zero very quickly. The motion is not a periodic motion. The variation of displacement x (t) with time T is shown below.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-29.png\")
<\/p>\n
\nThe energy of an undamped oscillator, E = \\(\\frac{1}{2}\\)kA\u00b2.
\nwhere A is the amplitude of an undamped oscillator. But for the damped oscillator, amplitude = Ae-bt\/2m<\/sup>.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-30.png\")
<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-31.png\")
<\/p>\n
\nSmall damping means that the dimensionless ratio
\n\\((b \/ \\sqrt{k m})\\) is much less than 1.<\/p>\n
\nFree oscillation:
\nWhen a body oscillates in the absence of external forces (eg. friction etc.) the oscillations are said to be free oscillations.<\/p>\n
\nWhen an external periodic force is applied to a damped harmonic oscillator, the oscillator will vibrate with a constant amplitude and frequency of vibration will be that of the applied periodic force. This type of oscillation is called forced oscillation.<\/p>\n
\nLet F(t) = F0<\/sub> cos\u03c9d<\/sub>t is an external force applied to a damped oscillator.<\/p>\n
\nF = – bv – kx + F0<\/sub> cos\u03c9d<\/sub>t
\nwhere -bv is the damping force and -kx is the linear restoring force.
\nF + bv + kx = F0<\/sub> cos\u03c9d<\/sub>t
\n\\(m \\frac{d^{2} x}{d t^{2}}+b \\frac{d x}{d t}+k x=F_{0} \\cos \\omega_{d} t\\)
\nThis is the differential equation of an oscillator of mass m on which a periodic force of (angular) frequency \u03c9d<\/sub> is applied. The oscillator initially oscillates with its natural frequency w. When we apply the external periodic force, the oscillation with the natural frequency die out, and the body oscillates with the (angular) frequency of the external periodic force.<\/p>\n
\nThe solution (displacement) of the above differential equation of forced oscillator can be written as
\nx(t) = Acos(\u03c9d<\/sub>t + \u03a6)
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-32.png\")
\nwhere m is the mass of the particle v0<\/sub> and x0<\/sub> are the velocity and the displacement of the particle at time t = 0, (which is the moment when we apply the periodic force)<\/p>\n
\nWhen b = 0 (damping force = 0) and \u03c9 = \u03c9d<\/sub>,
\nwe get A = \\(\\frac{F_{0}}{0}\\)
\nA = \u221e
\nThis is an ideal case. This case never arises in a real situation as the damping is never perfectly zero.<\/p>\n
\n(Small damping, driving frequency far from natural frequency).
\nIn this \u03c9d<\/sub>b << m(\u03c9\u00b2 – \u03c9d<\/sub>\u00b2). Hence we can neglect \u03c9d<\/sub>b. Hence amplitude of oscillation can be written as.
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-33.png\")
<\/p>\n
\n(Small damping, driving frequency close to natural frequency).
\nIn this case m(\u03c9\u00b2 – \u03c9a<\/sub>\u00b2) << \u03c9d<\/sub>b. Hence we can neglect m (\u03c9\u00b2 – \u03c9a<\/sub>\u00b2). Hence amplitude of oscillation can be written as
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-34.png\")
<\/p>\n
\nWhen the frequency of the external periodic force is varied, it is found that the amplitude of the forced vibration increases and reaches a maximum value and then decreases.<\/p>\n
\n![\"Plus](\"https:\/\/www.aplustopper.com\/wp-content\/uploads\/2019\/08\/Plus-One-Physics-Notes-Chapter-14-Oscillations-35.png\")
<\/p>\n
\nAll mechanical structures have one or more natural frequencies, and if a structure is subjected to a strong external periodic driving force that matches one of these frequencies, the resulting oscillations of the structure may rupture it.<\/p>\nPlus One Physics Notes<\/a><\/h4>\n","protected":false},"excerpt":{"rendered":"