Explosions send blast waves from the centre, which consist of an initial positive pressure wave front followed by a phase of negative pressure which pulls objects back towards the detonation centre. For this question, consider the positive pressure phase of a blast wave as shown below. Typically, the blast wave last 100 ms, and reaches a max pressure (gage) of around 70 kPa. Gage Pressure P(t) P Positive Phase P = 70 kPa 0.1 Time (s) P(1) A nearby road sign (shown above on the right), is approximated by a massless rod supporting a uniform thin disk (viewed from the side) of mass m, radius r, and frontal area A. The centre of the disk is at a height L = 2r from the ground, and the ground connection is modelled as a torsional spring k. Assume that the pressure-time relation during the positive phase is approximated as: P1 P(t) = (e-at + 3e-bt - 6t), 0 t 0.1s 4 The moment of inertia of a thin disk about its central diameter is J = mr. a. (5 pts) For the relation described above, determine the equation of motion for the sign during the positive phase in terms of m, L, P (t), k, A, and g. Use 0 as your coordinate. 12 = MR2 I, == Area A b. (10 pts) If the sign is initially at rest and a = 10,b=25, m = 1.5 kg, r = 0.25 m, and k = 1 x 106 Nm, calculate the response (t) of the sign during the positive phase of the blast wave. Hint: To simplify calculations and algebra, work with dummy variables for the effective mass, stiffness, and coefficients of the particular solution rather than m, L, P, etc. c. (10 pts) Neglect the negative pressure phase and assume that P(t) = 0 after the positive phase (t> 0.1 s). Determine the value of 0 one second after the positive phase has ended. Ix - - MR