IGNOU BPHE-102 - Oscillations and Waves

1. a) i) An object is executing simple harmonic motion. Obtain expressions for its kinetic and potential energies. ii) A spring mass system is characterized by k = 15 Nm-1 and m = 0.5 kg. The system is oscillating with amplitude of 0.40 m. Obtain an expression for the velocity of the block as a function of displacement and calculate its value at x = 0.15 m. b) Consider a particle undergoing simple harmonic motion. The velocity of the particle at position x1 is v1 and velocity of the particle at position x2 is v2. Show that the ratio of time period (T) and amplitude (A) is: c) Establish the equation of motion of a damped oscillator. Solve it for a weakly damped oscillator and discuss the significance of the results. d) A body of mass 0.2 kg is suspended from a spring of force constant 80 Nm−1. 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) e) Consider N identical masses connected through identical springs of force constant k. The free ends of the coupled system are rigidly fixed at x = 0 and x = l. 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. 2. 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. b) i) A sound wave of frequency 400 Hz travels in air at a speed of 320 ms-1 . 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 h-1 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 y = 5 sin π (4.0t - 0.02 x) 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 1.3 × 10 -4 kg m-1 . A transverse wave is propagating on the string and is described by the equation: y(x,t) = 0.021sin (x + 30 t) where x and y are in meters and t is in seconds. Calculate the tension in the string. d) Standing waves are produced by superposition of the following waves: y1 (x, t) = 0.2sin π(t - 2x) and y2 (x, t) = 0.2sin π(t+2x) (i) Obtain the resultant displacement of the particle at x at time t. (ii) For what value of x will the displacement be zero at all times? (iii) What is the distance between two nearest values of x at which displacements are zero? Is this distance related to the wavelength of the standing wave? 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 ms-1 .