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.

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.

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.

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

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.

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.