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UNIT 1 Oscillatory and Wave motion periodic motion A motion that repeats itself after equal intervals of time is known as periodic motion. Examples of periodic motion are a tuning fork or motion of a pendulum if you analyze the motion you will find that the pendulum passes through the me...
UNIT 1 Oscillatory and Wave motion periodic motion A motion that repeats itself after equal intervals of time is known as periodic motion. Examples of periodic motion are a tuning fork or motion of a pendulum if you analyze the motion you will find that the pendulum passes through the mean position only after a definite interval of time. What is Oscillatory Motion? Oscillatory motion is defined as the to and fro motion of an object from its mean position. The ideal condition is that the object can be in oscillatory motion forever in the absence of friction but in the real world, this is not possible and the object has to settle into equilibrium. Examples of Oscillatory Motion Following are the examples of oscillatory motion: Oscillation of simple pendulum Vibrating strings of musical instruments is a mechanical example of oscillatory motion Movement of spring Alternating current is an electrical example of oscillatory motion Difference between Oscillatory Motion and Periodic Motion Periodic motion is defined as the motion that repeats itself after fixed intervals of time. This fixed interval of time is known as time period of the periodic motion. Examples of periodic motion are motion of hands of the clock, motion of planets around the sun etc. Oscillatory motion is defined as the to and fro motion of the body about its fixed position. Examples of oscillatory motion are vibrating strings, swinging of the swing etc. Simple Harmonic Motion What is Simple Harmonic Motion? Simple Harmonic Motion or SHM is defined as a motion in which the restoring force is directly proportional to the displacement of the body from its mean position. The direction of this restoring force is always towards the mean position. For any simple mechanical harmonic system (system of the weight hung by the spring to the wall) that is displaced from its equilibrium position, a restoring force which obeys the Hooke’s law is required to restore the system back to equilibrium. Following is the mathematical representation of restoring force: The restoring force is a force which acts to bring a body to its equilibrium position. The restoring force is a function only of position of the mass or particle, and it is always F=−kx directed back toward the equilibrium position of the system. Where, F is the restoring elastic force exerted by the spring (N) k is the spring constant (Nm-1) x is the displacement from equilibrium position (m) Mass – Spring system: A torsional pendulum, or torsional oscillator, consists of a disk-like mass suspended from a thin rod or wire. When the mass is twisted about the axis of the wire, the wire exerts a torque on the mass, tending to rotate it back to its original position. The disc is twisted and released, it will undergo simple harmonic motion , provided the torque in the rod is proportional to the angle of twist. Figure shows an angular version of a simple harmonic osllator Damped harmonic oscillation: If the magnitude of the velocity is small, meaning the mass oscillates slowly, the damping force is proportional to the velocity and acts against the direction of motion (FD=−b). The net force on the mass is therefore The angular frequency for damped harmonic motion becomes Forced oscillations and resonance The paddle ball on its rubber band moves in response to the finger supporting it. If the finger moves with the natural frequency f0 of the ball on the rubber band, then a resonance is achieved, and the amplitude of the ball’s oscillations increases dramatically. At higher and lower driving frequencies, energy is transferred to the ball less efficiently, and it responds with lower-amplitude oscillations. Amplitude of a harmonic oscillator as a function of the frequency of the driving force. The curves represent the same oscillator with the same natural frequency but with different amounts of damping. Resonance occurs when the driving frequency equals the natural frequency, and the greatest response is for the least amount of damping. The narrowest response is also for the least damping. graph of the amplitude of a damped harmonic oscillator as a function of the frequency of the periodic force driving it. Each of the three curves on the graph represents a different amount of damping. All three curves peak at the point where the frequency of the driving force equals the natural frequency of the harmonic oscillator. The highest peak, or greatest response, is for the least amount of damping, because less energy is removed by the damping force. Note that since the amplitude grows as the damping decreases, taking this to the limit where there is no damping (b = 0), the amplitude becomes infinite. WAVES Progressive Waves Waves move energy from one place to another. What is Wavenumber? In physics, the wavenumber is also known as propagation number is defined as the number of wavelengths per unit distance. It is represented by k and the mathematical representation is given as follows: k=1/λ Where, k is the wavenumber λ is the wavelength Wavenumber Formula Wavenumber equation is mathematically expressed as the number of the complete cycle of a wave over its wavelength, given by – k=2π/λ Where, k is the wavenumber 𝜆 is the wavelength of the wave Measure using rad/m Superposition of waves The principle of superposition states that when two or more waves meet at a point, the resultant displacement at that point is equal to the sum of the displacements of the individual waves at that point. PHASE AND PHASE DIFFERENCE As in SHM the phase of a wave motion is given by the quantity which gives complete information about the wave at any instant and at any position. The argument of sine function is called Phase. Φ = 2π(t/T-x/λ) Phase Phase Difference
