Question

2. Expressing Oscillations Mathematically: A simple harmonic differential equation of motion mik), applies to the following systems. General solutions can be expressed in A, B, C or D forms and use any parameters listed in the question. (a) Consider a torsional pendulum, 16--kθ-τ where is the moment of intertia and τ is the torque. What is 60) and the characterictic frequency oo? (b) An air-filled flask with a cyclindrical shape neck of length 1 and cross-sectional area A can have oscillations in the 'slug' of the air in the cyclindrical section (like an ideal gas piston moving back and forth with displacement dx). The air in this air-piston has mass plA, where ρ is the density of the air. The pressure change in the piston obeys the ideal gas law of an adiabatic relation, i.e. PV-C dV Adx i. Show that the corresponding differential equation is dP=-7P-=-P Show that the a characteristic frequency of this oscillation piston is given by . Check that the units on co make sense. 0 Write a general solution of the equation of motion for the slug displacement oscillation, x(0), leave the amplitudes as unknowns.[ABCD forms] iii.

Below is A, B, C, D Form

Answer 1

m x^dot dot = -kx d^2 x/dt^2 = -k/m x d^2 x/dt^2 + k/m x = 0 (D^2 + k/m) x = 0 Auxillary equation: s^2 + k/m = 0 s = plusminus I squareroot k/m Homogenous solution: c_1 e^i squareroot k/m x^+ + C_2 e^i squareroot k/m x^+ let k/m = w_0^2 implies w_0 = squareroot k/m A: C e^i w_0 t + c^* e^-i w_0 t A: C e^i w_0 t + c^* e^-i w_0 t e^i w_0 t = i sin w_0 t + cos w_0 t e^i w_0 t = cos w_0 t - i sin w_0 t B: B_p cos w_0 t + B_q sin w_0 t D: A cos (w_0 t + phi) [using cos (x + y) = cos x co y = sin x sin y = cos x cos y - sin x sin y PV^r = C d^p V^r + Pr v^r - 1 dv = 0 dP/dV =

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