Periodic and Oscillatory Motion
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Questions and Answers

What is defined as periodic motion?

  • Motion that repeats after a regular interval of time (correct)
  • Motion that occurs only once
  • Motion that occurs without a defined pattern
  • Motion that varies randomly over time

    • Which type of oscillatory motion occurs in a U-tube?

  • Non-mechanical oscillation
  • Circular oscillation
  • Mechanical oscillation
  • Linear oscillation (correct)

    • What common feature is present in all types of oscillatory motion?

  • Uniform acceleration during the motion
  • A restoring force that increases with displacement (correct)
  • A constant speed throughout the motion
  • Lack of equilibrium position

    • Which of the following is NOT an example of a mechanical oscillatory system?

    <p>Earth's rotation around the sun</p> Signup and view all the answers

    In mechanical oscillatory systems, what two factors are primarily responsible for oscillation?

    <p>Inertia and restoring force</p> Signup and view all the answers

    What defines the equilibrium position in oscillatory motion?

    <p>The point where the restoring force is zero</p> Signup and view all the answers

    Which example demonstrates circular oscillation?

    <p>A solid sphere rolling in a cylinder</p> Signup and view all the answers

    What kind of oscillatory systems do not involve mechanical movement?

    <p>Electromagnetic oscillatory systems</p> Signup and view all the answers

    What is the nature of a non-mechanical oscillatory system?

    <p>The body's position remains constant while its properties vary.</p> Signup and view all the answers

    Which of the following is an example of an oscillatory motion?

    <p>Mass spring system</p> Signup and view all the answers

    What does the restoring force in simple harmonic motion depend on?

    <p>The displacement from the equilibrium position</p> Signup and view all the answers

    In the equation F = -kx, what does the constant 'k' represent?

    <p>Proportionality constant called force constant</p> Signup and view all the answers

    What is the general form of the differential equation for simple harmonic motion?

    <p>ω²x = 0</p> Signup and view all the answers

    Which of the following describes the angular frequency in simple harmonic motion?

    <p>The number of complete cycles per unit time</p> Signup and view all the answers

    What is the significance of the negative sign in the equation F = -kx?

    <p>It represents that the restoring force acts opposite to displacement.</p> Signup and view all the answers

    What does the equation x = A cos(ωt + φ) represent in the context of simple harmonic motion?

    <p>The position of the oscillator as a function of time</p> Signup and view all the answers

    What does the decrement of an oscillator measure?

    <p>Rate at which amplitude decreases</p> Signup and view all the answers

    In the context of underdamped oscillations, what does the symbol $\beta$ represent?

    <p>Decay constant</p> Signup and view all the answers

    What form does the velocity of an underdamped oscillation take according to the provided equations?

    <p>$v = re^{-\beta t} [\beta \cos(\omega_1 t + \theta) + \omega_1 \sin(\omega_1 t + \theta)]$</p> Signup and view all the answers

    Which equation defines the total energy of an oscillator?

    <p>$T.E. = K.E. + P.E.$</p> Signup and view all the answers

    What does the relaxation time indicate in damped oscillations?

    <p>Time for energy to decay to 1/e of its initial value</p> Signup and view all the answers

    What is the logarithmic decrement of the oscillator denoted by?

    <p>$\lambda$</p> Signup and view all the answers

    In over-damped oscillation, what is true about the relationship between $\beta$ and $\omega_0$?

    <p>$\beta^2 &gt; \omega_0^2$</p> Signup and view all the answers

    What is the formula for calculating kinetic energy in oscillation?

    <p>$K.E. = mv^2$</p> Signup and view all the answers

    What occurs when the dimension of the obstacle is comparable to the wavelength of the incident light?

    <p>Light bends at the edge of the obstacle.</p> Signup and view all the answers

    Which type of diffraction occurs when both the source and screen are at infinite distances from the obstacle?

    <p>Fraunhoffer Diffraction</p> Signup and view all the answers

    What type of wavefront is associated with Fresnel Diffraction?

    <p>Cylindrical or spherical wavefront</p> Signup and view all the answers

    What causes the differences in brightness observed in the diffraction pattern?

    <p>Destructive and constructive interference of light waves.</p> Signup and view all the answers

    In Fraunhoffer Diffraction, what is a characteristic of the rays used?

    <p>They are parallel and converged using a lens.</p> Signup and view all the answers

    What is the path difference at the angle θ for Fraunhoffer Diffraction at a single slit?

    <p>$e sinθ$</p> Signup and view all the answers

    Which condition is true for Fresnel Diffraction compared to Fraunhoffer Diffraction?

    <p>No lenses are utilized.</p> Signup and view all the answers

    How is the central bright image formed in Fraunhoffer Diffraction due to a single slit?

    <p>By converging rays at a point on screen.</p> Signup and view all the answers

    What is the condition for obtaining central or principal maxima in the context of wave interference?

    <p>When $sin(\alpha) = 0$</p> Signup and view all the answers

    For which values of $m$ does the condition for minima hold?

    <p>m = 1, 2, 3, 4, ...</p> Signup and view all the answers

    What is the relationship between intensity $I$ and amplitude $R$ in the given wave context?

    <p>I is equal to $KR^2$</p> Signup and view all the answers

    What approximations are made regarding $\alpha$ and $n$ in the formulation provided?

    <p>$\alpha$ is small and $n$ is large</p> Signup and view all the answers

    What expression represents the resultant amplitude $R$ derived from the superposition of waves?

    <p>$R = a sin(\alpha)$</p> Signup and view all the answers

    Which equation represents the phase difference related to the sine of the angle?

    <p>$Avg.phase = \frac{e sin(\theta)}{n}\cdot 2\pi$</p> Signup and view all the answers

    How can the intensity $I_0$ be expressed in terms of constant $K$ and amplitude $A$?

    <p>$I_0 = KA^2$</p> Signup and view all the answers

    At what position will the principal maxima be obtained?

    <p>At $\theta = 0$</p> Signup and view all the answers

    Study Notes

    Periodic and Oscillatory Motion

    • A motion that repeats itself after a fixed interval of time is called periodic motion.
    • Oscillatory motion is a specific type of periodic motion where an object moves back and forth around a central point called the equilibrium position.
    • The movement of the object is driven by a restoring force that pushes or pulls it back towards the equilibrium position.
    • This force increases as the object moves further away from the equilibrium position.
    • Oscillatory motion can be linear or circular.

    Types of Oscillatory Motion

    • Examples of linear oscillation include:

      • Mass-spring systems: A mass attached to a spring oscillates back and forth.
      • Fluid column in a U-tube: A column of fluid in a U-shaped tube oscillates up and down.
      • Floating cylinder: A cylinder floating in a liquid oscillates vertically.
      • Body dropped in a tunnel along earth diameter: A body dropped through a tunnel passing through the center of the earth would oscillate between the surface and the opposite side of the earth.
      • Strings of musical instruments: Vibrating strings on musical instruments produce sound through oscillations.

    • Examples of circular oscillation include:

      • Simple pendulum: A mass hanging from a string swings back and forth.
      • Solid sphere in a cylinder: A solid sphere rolling without slipping inside a cylinder oscillates in a circular path.
      • Circular ring suspended on a nail: A ring suspended on a nail oscillates in a circular motion.
      • Balance wheel of a clock: The balance wheel in a mechanical clock oscillates to regulate timekeeping.
      • Rotation of the earth around the sun: The Earth's orbit around the sun is a type of circular oscillation.

    Oscillatory Systems

    • An oscillatory system is composed of an object that exhibits to-and-fro motion around an equilibrium point due to a restoring force.
    • Oscillatory systems can be mechanical or non-mechanical.
    • Mechanical Oscillatory Systems: In this type, the object itself changes its position. Examples include the oscillations of a mass-spring system, a pendulum, and a body dropped through a tunnel.
    • Non-mechanical Oscillatory Systems: These involve periodic variations in a physical property rather than the position of an object. For example, variations in electric current in an oscillating circuit or fluctuations in pressure in a medium carrying sound waves.

    Simple Harmonic Motion (SHM)

    • This is the simplest type of oscillatory motion.
    • It occurs when the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction.
    • The mathematical formula for SHM restoring force is: F=-kx, where F is the restoring force, x is the displacement, and k is the force constant.
    • SHM is described by a sinusoidal function. This sinusoidal motion can be expressed as x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.
    • The angular frequency is related to the period (T) and frequency (f) of the motion by the following equations: ω = 2πf = 2π/T.

    Damped Oscillations

    • Real oscillations often experience damping, where the amplitude decreases over time.
    • This decrease is caused by energy dissipation due to factors like friction or air resistance.
    • The damped oscillation can be represented by the equation x(t) = Ae^(-βt) cos(ω1t + φ), where β is the damping coefficient and ω1 is the damped angular frequency.
    • Underdamped Oscillations: For a damping coefficient β that is less than the natural angular frequency ω0, the system oscillates with gradually decreasing amplitude.
    • Overdamped Oscillations: When β is greater than ω0, the system returns to equilibrium without oscillating.
    • Critically Damped Oscillations: In this case, β equals ω0, and the system returns to equilibrium in the shortest possible time without oscillating.

    Diffraction of Light

    • Light bends when it passes around an obstacle, particularly when the size of the obstruction is comparable to the wavelength of the light.
    • This phenomenon, called Diffraction, results in the spreading of light into the geometric shadow of the obstacle.
    • Diffraction can be classified as Fresnel diffraction or Fraunhofer diffraction.

    Fresnel Diffraction

    • Occurs when either the source of light, the screen, or both are a finite distance from the obstacle.
    • No lenses are used to make the rays converge or parallel.
    • The incident wavefront can be cylindrical or spherical.

    Fraunhofer Diffraction

    • This occurs when the light source and the screen are at an infinite distance from the obstacle.
    • Lenses are used to make the incident light rays parallel and then converge the diffracted rays on the screen.
    • The incident wavefront is planar.
    • Examples of Fresnel Diffraction: Diffraction at the narrow, straight edge of an obstacle
    • Examples of Fraunhofer Diffraction: Diffraction due to a single slit.

    Fraunhofer Diffraction Due to a Single Slit

    • When a beam of light passes through a narrow slit, the light diffracts, creating an interference pattern of alternating bright and dark bands on a screen.
    • The central bright fringe is the most intense and is located directly opposite the slit.
    • The position of the minima (dark bands) is determined by the formula e sin(θ) = mλ, where e is the slit width, θ is the angle of diffraction, m is the order of the minima (1, 2, 3, ...), and λ is the wavelength of light.
    • The position of the maxima (bright bands) is determined by the formula e sin(θ) = (m + 1/2)λ.

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