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 opposite to velocity.
m a = m d² x / d t² = - k x - b v
d² x /d t² = -k/m * x - b/m * dx/dt --- (2)
This is the equation of motion in the form of differential equation.
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A harmonic wave on a rope is described by
i) Calculate the wavelength and time period of the wave. ii) Determine the
displacement and acceleration of the element of the rope located at x = 0.58 m at
time, t = 0.41 s.
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
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.
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
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.
An ambulance siren has frequency 250 Hz. The ambulance is headed towards an
accident site with a speed of 90 km h−1, Two police officers on separate motor
cycles head for the same accident site: one follows the ambulance with a speed of
80 km h−1 and the other approaches the accident site from the other direction with a
speed of 80 km h−1. What frequency does ambulance siren has for each of the police
officers? Take the speed of sound equal to 340 ms−1.
What do you understand by the normal modes of a coupled oscillator? If a coupled
system has many normal modes, do all normal modes have the same frequency? In a
solid, the speed of elastic longitudinal wave is 1.35 ms−1 If the Young’s modulus of
elasticity of the solid is 2 x 1011 N m −2 ,calculate its mass density.