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OSCILLATIONS, , Chapter-14, , OSCILLATIONS, , Periodic motion: A motion that repeats itself at regular intervals of time is called periodic motion., Ex: Motion of planets in solar system, uniform circular motion., Oscillatory motion: A motion in which a body moves to and fro between two extreme positions, about an equilibrium position., Ex: boat tossing up and down, piston of a steam engine, motion of simple pendulum., Equilibrium (Mean) position: It is the position of a body during oscillatory motion at which the, net external force acting on the body is zero., It is the position, at which if it is at rest, it remains at rest forever., Oscillations or vibrations: The motion of a body between two extreme positions forms oscillations, or vibrations., Note: (i) There is no significant difference between oscillations and vibrations. When the frequency, is small we call it oscillation, while the frequency is high we call it vibrations., (ii) Every oscillatory motion is periodic; but every periodic motion need not be oscillatory., Importance of oscillatory motion: This motion is basic to physics. In musical instruments we come, across vibrating strings, membranes in drums and diaphragms in telephone and speaker system, vibrate, vibrations of air molecule, vibrations of atoms in solid include oscillatory motion. The, concepts of oscillatory motion are required to understand many physical phenomena listed above., Description of oscillatory motion: The description of oscillatory motion requires some, fundamental concepts like period, frequency, displacement, amplitude and phase., Period or Time period (T): The smallest interval of time after which a periodic motion repeats is, called period., In case of oscillation, the time taken by the body to complete one oscillation is called period. SI unit, of period is, ., Frequency ( ): Number of times a periodic motion repeats per unit time is called frequency., In case of oscillations, number of oscillations per unit time is called frequency. SI unit of frequency, is, ., oscillation per second., Note: Relation between period and frequency is given by,, , ⁄ or, , ⁄, , Displacement (x or y): The term displacement refers to change of physical quantity with time. In, periodic motion displacement may be linear as well as angular., Linear displacement: The straight line distance travelled by a particle, from its equilibrium position., Angular displacement: It is the angle through which position vector of the body rotates in a given, time., Karnataka PUC Physics Telegram channel, , Page | 1

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OSCILLATIONS, , Amplitude (A): The maximum displacement of the particle from its equilibrium, position is called amplitude., Periodic function: Any function which repeats itself after a regular interval of time, is called periodic function., In periodic motion displacement is periodic function and it can be represented by a mathematical, function of time. The simplest of these functions is given by,, ., If, is increased by an integral multiple of, radian, the value of the function remains same and, is periodic., , (, , ), , Note: (i) In, , the term is called angular frequency., (ii) The function, is also periodic., (iii) The linear combination of both sine and cosine function is also periodic and it is represented, by, and it is called Fourier series., By putting,, and, ( ), , √, , Simple harmonic motion (SHM): The oscillatory motion is said to be simple harmonic, if the, displacement of the particle from the origin varies with time as;, or, ., Simple harmonic motion is a periodic motion in which displacement is a sinusoidal function of, time., Note: The simplest kind of periodic motion is simple harmonic motion., Analysis of simple harmonic motion:, Consider a particle oscillating back and forth about the origin along, and, as shown., , Figure shows graph of, , between the limits, , versus which gives the values of displacements as function of time., , Phase: During the periodic motion, the position and velocity of the particle at any time is, determined by the term, in cosine function. This quantity is called phase of the motion., Phase constant (Phase angle):, The value of phase at, is and it is called the phase constant or phase angle., Karnataka PUC Physics Telegram channel, , Page | 2

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OSCILLATIONS, , Simple harmonic motion and uniform circular motion:, Consider a particle moving with a uniform sped along the circumference of circle of radius ., Let the particle start from the point with a constant speed ., After some time it reaches to ., Draw, perpendicular to, ., represents the projection of position vector of the particle on, ., When the particle moves from, to its projection of the position, vector moves from, to . As the particle moves from to , its, projection moves from to . Similarly the particle moves from, to, via , its projection moves from to, and, to . This shows, that if the particle moves uniformly on a circle, its projection on the, diameter (, ) of the circle executes SHM., The position of the particle on the circle is given by,, The displacement of the projection on, is given by,, with same amplitude but different in phase by ⁄ ., , which is also SHM, , Equation of SHM:, Consider a particle moving on a circle of radius with uniform velocity ., Let the particle start from and subtend an angle in time and reaches ., , The projection of the particle on, , is,, , If the particle starts from ,, Velocity of the particle:, We have,, , Further,, , √, √, , ( ), , √, √, Negative sign shows that, , has a direction opposite to the positive direction of, , ., , The above equation tells that,, (i) When, ,, - velocity is maximum, velocity is maximum at equilibrium (mean), position, (ii) When, ,, - velocity is minimum, velocity is minimum at extreme position., Karnataka PUC Physics Telegram channel, , Page | 3

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OSCILLATIONS, , Acceleration:, We have, , Negative sign indicates that the direction of displacement and acceleration are opposite to each, other., (i) When, ,, , acceleration is minimum at mean position., (ii) when, ,| |, , acceleration is maximum at extreme position., , Force law for SHM:, Acceleration of a particle executing SHM is given by,, From Newton’s second law,, , Negative sign indicates that force and displacement are oppositely directed., Note: (i) A particle oscillating under a force given by, , is called linear harmonic oscillator, , √, , Energy in SHM:, A particle executing SHM possess,, (i) Kinetic energy - because it is moving., (ii) Potential energy - because it is subjected to conservative force, , [, , Karnataka PUC Physics Telegram channel, , ], , Page | 4

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OSCILLATIONS, , Variation of Kinetic energy and potential energy of oscillator:, , At mean position kinetic energy is maximum and potential energy is zero., , At the extreme positions kinetic energy is zero and potential energy is maximum., , Some systems executing SHM:, There are no practical examples for absolutely pure simple harmonic motion. But under certain, conditions, some systems can be considered as approximately simple harmonic., (i) Oscillations due to spring (Expression for Time period of oscillating string):, Consider a block of mass attached to a spring. The other end of the spring is rigidly fixed., If the block is pulled and released, it executes to and fro motion., Let, be the mean position of the block., The restoring force of the block is given by,, ( ), The standard equation for SHM is,, , √, √, (ii) Simple pendulum (Expression for time period of Simple pendulum):, Consider a simple pendulum of mass tied to a string of length ., Let the bob is set into oscillations., Let be the position of the bob at time ., Let be the angle made by the string with the vertical., Force acting on the bob are,, (i) weight of the bob, , vertically downwards, which can be, resolved into two components;, along the string and, perpendicular to the string., (ii) Tension in the string towards point of suspension., The bob has two accelerations (i) radial acceleration, (ii) tangential acceleration, Radial acceleration provided by, Tangential acceleration provided by, Karnataka PUC Physics Telegram channel, , Page | 5

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OSCILLATIONS, , Radial force gives zero torque., | ⃗|, , |⃗, , ⃗|, , Negative sign indicates that the restoring torque tends to reduce angular displacement., By Newton’s second law,, , Comparing with,, , √, , √, , √, , √, , √, , Damped simple harmonic motion:, Damped oscillations: The oscillations of decreasing amplitude due to force opposing the motion, of the particle are called damped oscillations., In damped oscillations, the energy of the system dissipates continuously., Damping force: The force which opposes the simple harmonic motion of the particle is called, damping force. Ex: air drag, viscous force., Analysis of Damped oscillations:, Generally damping forces are velocity dependent., The damping force, is given by,, where is the positive constant called damping co-efficient and depends on characteristics of the, medium. The negative sign indicates the, is opposite to ., The restoring force acting on the object is,, The net force on the oscillator is,, , This is the differential equation of damped SHM., Karnataka PUC Physics Telegram channel, , Page | 6

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OSCILLATIONS, , (, √, , (, , ), , ), , The amplitude of the oscillation decreases exponentially with time and finally becomes zero., Energy of the damped oscillator:, , (, (, (, , ), , ), ), , The equation shows that the total energy of the system decreases exponentially with time., Free oscillations: When a particle set into oscillations, it oscillates with its own frequency, oscillations are called free oscillations and the frequency is called natural frequency., All the free oscillations eventually die out because of the ever present damping forces., , , these, , Forced or driven oscillations: When a body is subjected to periodic force, it oscillates with the, frequency of the periodic force. Such oscillations are called forced oscillations., Note: In forced oscillations the system oscillates not with its natural frequency but at the frequency, of the external agency., Analysis of forced oscillations:, , The particle initially oscillates with natural frequency . The oscillations with natural frequency, die out due to damping. Then the particle oscillates with frequency of the external periodic force., Displacement is given by,, [, (, , ], , ⁄, , ), , where, is mass of the particle,, is velocity and, at which we apply periodic force., , is the displacement of the particle at time, , Case(i) Small damping, Applied frequency far from natural frequency., (, [, , ., , ), ], , ⁄, , Karnataka PUC Physics Telegram channel, , Page | 7

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OSCILLATIONS, , If we go on changing the applied frequency, the amplitude tends to infinity, when, This cannot happen in reality as zero damping is ideal., Case(ii) Applied frequency close to natural frequency., When, , ,, , (, , ), , The maximum amplitude for a given applied frequency depends on the applied frequency and the, damping force., Resonance: The phenomenon of increase in amplitude when applied frequency is close to the, natural frequency of the oscillator is called resonance., The frequency at which resonance takes place is called resonant frequency., , Karnataka PUC Physics Telegram channel, , Page | 8

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OSCILLATIONS, , Suggested Questions., One mark., 1) What is frequency of an oscillator?, 2) Define Amplitude of simple harmonic oscillator., 3) What is phase of an oscillating particle?, 4) What happens to the time period of simple pendulum when it is taken from equator to the, pole?, 5) What are damped oscillations?, 6) What is resonance?, Two marks., 1) On an average the human heart is found to beat 75 times in minute. Calculate its frequency., 2) Define SHM with an example., 3) What is simple harmonic motion? Write an expression for linear oscillator., 4) Mention any two characteristics of SHM., 5) Mention the expression for the velocity of the particle executing SHM and explain the symbol., 6) Write the expression for total energy of a simple harmonic motion and explain the symbols., 7) Where is the velocity of the body maximum and minimum in case of simple harmonic, motion?, 8) Mention an expression for the period of oscillation of a spring and explain the terms., 9) What are forced oscillations? Mention the condition for resonance., Three marks., 1) Mention the expression for the velocity of the particle executing SHM and mention the, position where it is maximum and minimum?, 2) Mention the expression for the acceleration of the particle executing SHM and mention the, position where it is maximum and minimum?, 3) Obtain an expression for Time period of oscillating string., Five marks., 1) What is SHM? Write its characteristics and give its graphical representation., 2) Obtain an expression for kinetic and potential energies of a particle in SHM varying between, zero and their maximum values with diagram giving total energy., 3) Derive the expression for time period of simple pendulum., Numerical Problems., 1) The equation of a sinusoidal wave travelling along negative x-axis is, where, and, are in meters and is in seconds. Calculate the amplitude, wave length,, frequency and wave velocity., 2) The displacement of an oscillating particle varies with time according to the equation, where is in metre and time in second. Calculate, (a) Amplitude of oscillation (b), Time period of oscillation (c) Maximum velocity of wave particles and (d) Acceleration of the, wave., 3) A particle executes SHM along the x-axis, its displacement varies with the time according to, the equation:, , where, , in metre and, , is in second. Determine the, , amplitude, frequency, period and initial phase of the motion., , Karnataka PUC Physics Telegram channel, , Page | 9