Answer to Question #26258 in Mechanics | Relativity for zameer

Question #26258
a) Write the equation of motion of a simple harmonic oscilla tor which has an amplitude of 5 cm and it executes 150 oscillations in 5 minutes with an init ial phase of 45 ° . Also obtain the value of its maximum velocity. (3+2) b) If the displacement of a particle executing SHM be 10 cm and 12 cm when the corresponding velocities are 16 cms − 1 and 14 cms − 1 respectively, calculate the amplitude of motion. (5) c) Derive expressions for potential energy and kinetic energy of an oscillating spring-mass system. (5+5) d) Discuss the principle of superposition. Two collinear SH Ms, with amplitudes 5 cm and 12 cm are superposed. Calculate the resultant amplitude when the SHMs differ in phase by (i) 60 ° , (ii) 90 ° and (iii) 120 ° . (4+6) e) Establish the differential equation for a damped oscillat or. Show that, for weak damping, the solution of the differential equation for the damped oscillat or is given by ) ( cos ) ( exp ) ( 0 φ + ω − = t bt a t x d (4+6) f) What do you understand by weakly damped forced oscillator and it s transient and steady states? Show that the average power absorbed by a forced oscill ator is given by ] 4 ) [( 2 2 2 2 2 0 2 2 0 ω + ω − ω ω         = > < b m bF P (5+5) g) What do you understand by the normal modes of coupled oscillators ? If a coupled system has many normal modes, do all normal modes have the sa me frequency? Calculate the velocity of elastic longitudinal wave along a stretched steel wire, given density of steel = 8000 kgm − 3 , Young’s modulus of elasticity = 2 × 10 11 Nm − 2 . (3+2+5) 2. a) Two points x 1 and x 2 at x = 0 and x = 1 m are observed. The transverse motion of the two points are found to be as follows: t t x y π = 3 sin 2.0 ) , ( 1 and       π + π = 8 3 sin 2.0 ) , ( 2 t t x y Calculate the frequency, wavelength and speed of the wave. (5) 4 b) A sinusoidal wave is described by cm ) 95.5 20.4( sin 0.4 ) , ( t x t x y − = where x is the position along the wave propagation. Determine th e amplitude, wave number, wavelength, frequency and velocity of the wave. ( 2 × 5 = 10) c) Two waves, travelling along the same direction, are given by ) ( sin ) , ( 1 1 1 x k t a t x y − ω = and ) ( sin ) , ( 2 2 2 x k t a t x y − ω = Suppose that ω 1 and k 1 are respectively slightly greater than ω 2 and k 2 . (i) Derive an expression for the resultant wave obtained by their super position. (ii) Explain the formation of wave packet. (5+5) d) Standing waves are produced by the superposition of two waves given by m ) 2 ( sin 2.0 ) , ( 1 x t t x y − π = and m ) 2 ( sin 2.0 ) , ( 2 x t t x y + π = (i) Obtain the resultant displacement of the particle at x at time t . (ii) At what value of x is displacement zero at all times. (iii) What is the distance between two nearest values of x at which the displacement is zero. Is it related to wavelength ? (3+3+4) e) A 125 cm long string has a mass 2.0 g and it is str etched with a tension of 7.0 N between fixed supports. What is the speed of the transverse wave on the string?
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