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Oscillations And Waves

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Oscillations and Waves PDF Notes, Important Questions and Synopsis

SYNOPSIS
  • Periodic motion: Motion of an object which regularly returns to a given position after a fixed time interval.
  • Oscillation:
    A special type of periodic motion in which a particle moves to and fro about a fixed point.
  • Periodic motion may be oscillatory or non-oscillatory:
    Motion of a planet around the sun is periodic but not oscillatory.
  • Amplitude: Maximum displacement of a particle from the mean position.
  • Time period: Minimum time taken by a particle to repeat its motion.
  • Basic differential equation of SHM:

    begin mathsize 12px style fraction numerator straight d squared straight y over denominator dt squared end fraction plus straight omega squared straight y equals 0 end style
  • Equation of a particle performing SHM

    begin mathsize 12px style straight y equals Asin open parentheses ωt plus straight ϕ close parentheses end style
  • Velocity of a particle:

    begin mathsize 12px style straight v equals straight omega square root of straight A squared minus straight y squared end root end style

    Velocity is maximum at the mean position and minimum at the extremes. 
    Maximum velocity ; minimum velocity v = 0

  • Acceleration of a particle:

    begin mathsize 12px style straight a equals negative straight omega squared straight y end style
    Acceleration is maximum at the extremes and minimum at the mean position.
    Maximum acceleration ; minimum acceleration a = 0

  • Phase difference:
    Same phase SHM
     Error converting from MathML to accessible text.
    Opposite phase SHM 
    begin mathsize 12px style straight ϕ equals straight pi comma 3 straight pi comma 5 straight pi. .. open parentheses 2 straight n plus 1 close parentheses straight pi end style
    Relation between phase difference and time difference 
    begin mathsize 12px style Δϕ equals ωΔt equals fraction numerator 2 straight pi over denominator straight T end fraction Δt end style

  • Energy of a body in SHM:
    Total energy of a particle in SHM consists of two parts—kinetic energy (KE) and potential energy (PE). The sum of PE and KE remains constant.
    At the mean position, kinetic energy is maximum and potential energy is minimum.
    At the extreme position, kinetic energy is zero and potential energy is maximum.
    Kinetic energy K
     begin mathsize 12px style equals 1 half mv squared equals 1 half mω squared open parentheses straight A squared minus straight x squared close parentheses equals 1 half mA squared straight omega squared Cos squared wt end style
    Potential energy U
     begin mathsize 12px style equals straight K over 2 straight x squared equals 1 half mω squared straight x squared equals 1 half mω squared straight A squared Sin squared ωt end style

    Total energy = K + U 
     begin mathsize 12px style 1 half mA squared straight omega squared Cos squared wt plus 1 half mω squared straight A squared Sin squared ωt equals 1 half mω squared straight A squared end style

  • Superposition of two SHMs:
    In the same direction and of the same frequency:
    begin mathsize 12px style table attributes columnalign left end attributes row cell straight x subscript 1 equals straight A subscript 1 sinwt end cell row cell straight x subscript 1 equals straight A subscript 2 sin open parentheses wt plus straight ϕ close parentheses end cell row cell straight x equals straight A subscript 1 sinwt plus straight A subscript 2 sin open parentheses wt plus straight ϕ close parentheses equals Asin open parentheses wt plus straight ϕ to the power of apostrophe close parentheses end cell row cell straight A equals square root of straight A subscript 1 squared plus straight A subscript 2 squared plus 2 straight A subscript 1 straight A subscript 2 cosϕ end root end cell row cell straight ϕ to the power of apostrophe equals tan to the power of negative 1 end exponent open square brackets fraction numerator straight A subscript 2 sinϕ over denominator straight A subscript 1 plus straight A subscript 2 cosϕ end fraction close square brackets end cell end table end style
    In the same direction but of different frequencies:

    begin mathsize 12px style straight x equals straight A subscript 1 sinω subscript 1 straight t plus straight A subscript 2 sinω subscript 2 straight t end style
    Resultant motion is not SHM.
    In two perpendicular directions,
    Case 1: If begin mathsize 12px style straight theta equals 0 text  or  end text straight pi end style, the path will be a straight line and the amplitude will be begin mathsize 12px style square root of straight A subscript 1 squared plus straight A subscript 2 squared end root end style
    Case 2: If begin mathsize 12px style straight theta equals text end text straight pi divided by 2 end style, the path will be an ellipse. The resultant motion is not SHM.

  • Spring mass system:
    Time period of the spring mass system oscillating in the horizontal plane =
     begin mathsize 12px style straight T equals 2 straight pi square root of straight m over straight k end root end style

    Time period of the spring mass system oscillating in the vertical plane =
     begin mathsize 12px style straight T equals 2 straight pi square root of straight m over straight k end root end style
    Series combination of springs
    begin mathsize 12px style 1 over straight k equals 1 over straight k subscript 1 plus 1 over straight k subscript 2 end style
    Parallel combination of springs
    begin mathsize 12px style straight k equals straight k subscript 1 plus straight k subscript 2 end style

    Spring cut into two parts in the ratio m:n

    begin mathsize 12px style straight k subscript 1 equals fraction numerator open parentheses straight m plus straight n close parentheses straight k over denominator straight m end fraction comma straight k subscript 2 equals fraction numerator open parentheses straight m plus straight n close parentheses straight k over denominator straight n end fraction end style

  • Oscillation of a two-particle system:

    Time period of the blocks
     begin mathsize 12px style straight T equals 2 straight pi square root of fraction numerator straight m subscript 1 straight m subscript 2 over denominator straight k open parentheses straight m subscript 1 plus straight m subscript 2 close parentheses end fraction end root equals 2 straight pi square root of straight mu over straight K end root end style
    Amplitude of the blocks

    begin mathsize 12px style table attributes columnalign left end attributes row cell straight A subscript 1 plus straight A subscript 2 equals straight x subscript 0 end cell row cell straight A subscript 1 equals fraction numerator straight m subscript 2 straight x subscript 0 over denominator straight m subscript 1 plus straight m subscript 2 end fraction end cell row cell straight A subscript 2 equals fraction numerator straight m subscript 1 straight x subscript 0 over denominator straight m subscript 1 plus straight m subscript 2 end fraction end cell end table end style

  • Simple pendulum:
    Time period
    begin mathsize 12px style straight T equals 2 straight pi square root of straight l over straight g end root end style
    Time period when the point of suspension is accelerating up=
    begin mathsize 12px style straight T equals 2 straight pi square root of fraction numerator straight l over denominator straight g plus straight a end fraction end root end style
    Time period when the point of suspension is accelerating down =
     begin mathsize 12px style straight T equals 2 straight pi square root of fraction numerator straight l over denominator straight g minus straight a end fraction end root end style
    Time period when the pendulum is accelerating horizontally =
     begin mathsize 12px style straight T equals 2 straight pi square root of fraction numerator straight l over denominator straight g squared plus straight a squared end fraction end root end style
    Time period when the pendulum is accelerating down an inclined plane =
     begin mathsize 12px style straight T equals 2 straight pi square root of straight l over gcosθ end root end style

  • Physical pendulum:
    begin mathsize 12px style straight T equals 2 straight pi square root of straight I over mgl end root end style

  • Oscillation of a floating body in a liquid:
    begin mathsize 12px style table attributes columnalign left end attributes row cell straight omega equals square root of gρ subscript straight l over hρ subscript straight s end root end cell row cell straight T equals 2 straight pi square root of hρ subscript straight s over gρ subscript straight l end root end cell end table end style
    begin mathsize 12px style straight rho subscript straight l end style =
     density of liquid
    begin mathsize 12px style straight rho subscript straight s end style = density of solid

  • Motion of a ball in a tunnel through the earth:
    Error converting from MathML to accessible text.

  • Sound waves:
    Sound is a mechanical and longitudinal wave created by a vibrating source. It needs a medium for its propagation. In a longitudinal wave, particles of the medium vibrate in the direction of the propagation of the wave.

  • In a transverse wave, particles of the medium oscillate along the direction of the propagation of the wave?

  • Wavelength of a progressive wave is the distance between two consecutive points of the same phase at a given time. In a stationary wave, it is twice the distance between two consecutive nodes or antinodes.

  • Speed of sound wave:
    Speed of a sound wave depends on elastic and inertial properties of the medium.

    begin mathsize 12px style straight v equals square root of fraction numerator text elastic property end text over denominator text inertial property end text end fraction end root end style

  • Speed of sound wave by Newton’s formula:
    According to Newton, when sound propagates, the temperature of the gas remains constant. The velocity of sound in air is 280 m/s.

  • Laplace’s correction: According to Laplace, heat of the medium remains constant instead of its temperature; therefore, he replaced isothermal elasticity by adiabatic elasticity.
     begin mathsize 12px style straight v equals square root of straight B subscript ad over straight rho end root equals 332 straight m divided by straight s end style

  • Factors affecting the speed of sound in a gas:
    Effect of pressure: With a change in pressure, density also changes, and hence, pressure has no effect on the speed of sound in a gas.
    Effect of temperature: begin mathsize 12px style straight v equals square root of γRT over straight M subscript straight o end root semicolon straight v proportional to square root of straight T end style, with begin mathsize 12px style 1 to the power of 0 straight C end style rise in temperature, the velocity of sound in air increases by 0.61 m/s.
    Effect of humidity: Speed of sound increases with increase in humidity.

  • Intensity of sound wave:
    Intensity decreases in proportion to the square of the distance from the source.

    begin mathsize 12px style straight I equals fraction numerator straight P over denominator 4 πr squared end fraction end style

  • Doppler effect:

    When a source of sound and a listener are in motion relative to each other, the frequency of the sound heard by the listener is not the same as source frequency. The general formula for frequency is begin mathsize 12px style straight f equals straight f subscript 0 open parentheses fraction numerator straight v plus-or-minus straight v subscript 0 over denominator straight v plus-or-minus straight v subscript 0 end fraction close parentheses end style. Frequency tends to increase when the source and observer move close to each other and decreases when the source and observer move away from each other.

  • Principle of superposition of waves:

    When two or more waves traverse the same medium, the displacement of any element of the medium is the algebraic sum of the displacement due to each wave.
    begin mathsize 12px style straight y equals sum from straight i equals 1 to straight n of straight f subscript straight i open parentheses straight x minus vt close parentheses end style

  • Organ pipe:
    Standing waves are the result of interference between longitudinal sound waves travelling in opposite directions. In a pipe open at both ends, the harmonics are given by begin mathsize 12px style fraction numerator nv over denominator 2 straight L end fraction end style , whereas for a pipe closed at one end, the harmonics are given by begin mathsize 12px style fraction numerator open parentheses 2 straight n plus 1 close parentheses straight v over denominator 4 straight L end fraction end style.

  • Interference of waves: 
    Constructive interference: The resultant amplitude is greater than the amplitude of individual waves.

    begin mathsize 12px style straight A subscript max equals straight A subscript 1 plus straight A subscript 2 end style

    Destructive interference: The resultant amplitude is less than the amplitude of individual waves.

    begin mathsize 12px style straight A subscript min equals straight A subscript 1 minus straight A subscript 2 end style

  • Reflection of waves at fixed end and free end:
    When a wave gets reflected from a fixed end, it suffers a phase change of π, i.e. it gets inverted.
    When a wave gets reflected from a free end, the pulse is not inverted, but the free end overshoots twice the amplitude.

  • Standing wave:
    The general equation of the standing wave is given by . In standing waves, every element of the medium oscillates in SHM with the same angular frequency ω. The amplitude of SHM of a given element depends on the location x of the element in the medium.

  • Characteristics of stationary waves:
    Disturbance does not move in any direction.
    Minimum displacement points are called nodes, and maximum displacement points are called antinodes.
    Amplitude of particles is different at different points.
    Particles in a particular segment between two nodes vibrate in the same phase, but particles in neighbouring nodes vibrate in the opposite phase.

  • Resonance: When the frequency of the applied force is equal to one of the natural frequencies of the system, the amplitude of the resulting motion is the greatest.

  • Beat: 
    Periodic variation in amplitude at a given point due to the superposition of two waves having slightly different frequencies.
    Beat frequency
     begin mathsize 12px style straight f subscript beat equals open vertical bar straight f subscript 1 minus straight f subscript 2 close vertical bar end style

  • Sonometer:
    A device used to measure the velocity of transverse mechanical waves.
    Velocity of a wave in a string is given by
     begin mathsize 12px style straight v equals square root of straight T over straight mu end root end style

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