Consider N identical masses connected through identical springs of force constant k. The free ends of the coupled system are rigidly fixed at and
The masses are made to execute longitudinal oscillations on a frictionless table.
i) Depict the equilibrium as well as instantaneous configurations.
ii) Write down their equations of motion, decouple them and obtain frequencies of normal modes.
We will numerically solve the problem step by step. --- ### Step 1: Equilibrium and Instantaneous Configurations - Consider \( N \) identical masses, each of mass \( m \), connected by \( N+1 \) identical springs of force constant \( k \). - The system is fixed at both ends, meaning the first and last springs are attached to rigid walls. - At equilibrium, the masses are evenly spaced along the \( x \)-axis. - _______ _________ ________ _____ ___ _____ ___ __________ _______ ______.
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e) i) Show that only odd harmonics can be generated in a closed-end organ pipe.
ii) Determine the fundamental frequency and the first 3 overtones of an organ pipe
of length 1.7 m and closed at one end. Take the speed of sound to be 340
b) Consider a particle undergoing simple harmonic motion. The velocity of the particle at position is
and velocity of the particle at position
is
Show that the ratio of time period (T) and amplitude (A) is:
i) A sound wave of frequency 400 Hz travels in air at a speed of 320 Calculate the phase difference between two points on the wave separated by a distance of 0.2 m along the direction of travel of the wave.
ii) A train moving with speed 72 km emits a whistle of frequency 500 Hz. A person is standing stationary on the platform. Calculate the frequency heard by the person if the train (i) approaches and (ii) recedes away from the listener
c) i) The equation of transverse wave on a string is given by
where y and x are in cm and t is in second. Calculate the maximum speed of a particle on the string and wavelength of the wave.
ii) The linear density of a vibrating string is A transverse wave is propagating on the string and is described by the equation:
sin
where x and y are in meters and t is in seconds. Calculate the tension in the string.
a) A Transverse waves propagating on a stretched string encounter another string of different characteristic impedance. (i) Write down the equations of particle displacement due to the incident, reflected and transmitted waves. (ii) Specify the boundary conditions and (iii) use these to obtain expressions for reflection and
transmission amplitude coefficients.
d) A body of mass 0.2 kg is suspended from a spring of force constant 80 A
damping force is acting on the system for which γ = 4 Nsm−1. Write down the equation of motion of the system and calculate the period of its oscillations. Now a harmonic force F =10cos10t is applied. Calculate a and θ when the steady state
response is given by a cos(ωt − θ).