What is the effect of damping in an oscillatory system? Differentiate between heavy and critical damping. Show that the displacement of a weakly damped oscillator is given by
x(t ) = a0 exp(−bt ) cos(ωd t − φ)
where symbols have their usual meanings.
Let us take a spring mass system with a natural frequency ω0.
m = mass of the body oscillating
Spring restoration force F = - k x , k = spring constant.
Damping force F_d = - b v , where b = damping factor.
x = displacement.
v = velocity of mass.
ωâ‚€ is the frequency of the spring mass when there is no damping. Only force is spring force.
so, m d² x / d t² = - k x has the solution:
x = A0 Sin (ωâ‚€ + Ф₀) --- (0)
and ωâ‚€ = √(k/m) or ω0² = k/m ---- (12)
===========================
net force acting on mass m is: F = - k x - b v --- (1)
The minus sign is because spring restoration force is opposite to displacement direction and the damping force is in the direction
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Two waves, travelling along the same direction, are given by
y1(x, t) = asin (ω1t − k1x)
and y2 (x, t) = asin (ω2t − k2x)
Suppose that the values of ω1 and k1 are respectively slightly greater than ω2 and
k2 . i) Obtain an expression for the resultant wave due to their superposition. ii)
Explain the formation of wave packet.
A sinusoidal wave is described by
y(x, t) = 3.0 sin (5.95t − 4.20x) cm
where x is the position along the wave propagation. Determine the amplitude, wave
number, wavelength, frequency and velocity of the wave.
The amplitude of an oscillator is 8 cm and it completes 100 oscillations in 80s.
i) Calculate its time period and angular frequency. ii) If the initial phase is π/4, write expressions for its displacement and velocity. iii) Calculate the values of maximum
velocity and acceleration
What is the effect of damping in an oscillatory system? Differentiate between heavy and critical damping. Show that the displacement of a weakly damped oscillator is given by
x(t ) = a0 exp(−bt ) cos(ωd t − φ)
where symbols have their usual meanings.
A body of mass 0.15 kg executes SHM described by the equation
x(t ) = 2 sin(πt + π / 4)
where x is in meters and t is in seconds.
i) Determine the amplitude and time period of the oscillation.
ii) Calculate the initial values of displacement and velocity.
iii) Calculate the values of time when the energy of the oscillator is purely kinetic.
The equation of motion of a damped harmonic oscillator is given by
with m = 0.25 kg, b = 0.14 s −1
and ω0 = 18.4 s −1.
Calculate i) the time period; ii) number of oscillations in which its amplitude will become half of its initial value; and iii) number of oscillations in which its
mechanical energy will reduce to half of its initial value.
Giving the necessary mathematical expressions, discuss the transient and steady state of a weakly damped forced oscillator. Show that the average power absorbed by a forced oscillator is given by
