A block of mass m is dropped from a height h above a spring k. as shown in Fig.

Question

A block of mass m is dropped from a height h above a spring k. as shown in Fig. P10.9 Beginning at the time of impact (t = 0). the position x(t) of the block satisfies the following second. order differential equation and initial conditions: Determine the transient solution xtran(t), and determine the frequency of oscillation. Determine the steady-state solution, xss(t). Determine the total solution x(t). subject to the initial conditions. Mark each of the following statements as true (T) or false (F): Increasing the stiffness k will increase the frequency of x(t). Increasing the height h will increase the frequency of x(t). Increasing the mass m will decrease the frequency of x(t). Doubling the height h will double the maximum value of x(t).

Explanation / Answer

x'' + w^2 x = g so Complimentary solution : (D^2 + w^2 )x =0 (D^2+w^2) =0 D = iw and D = - iw so x = Ae^(iwt) + Be^(-iwt) = C cos wt + B sin wt Particular solution x = g/(D^2+w^2) = (1+D^2/w^2)^(-1) * g/w^2 = mg/k so x (t) = C cos wt + B sin wt + mg/k at t =0 ; x =0 so C = - mg/k x'(t) = -Cw sin wt + Bw cos wt at t=0 ; x'(0) = sqrt (2*g*h) x'(0) = Bw = sqrt (2*g*h) B = sqrt(2*g*h) / w A) -mg/k * cos wt + sqrt(2*g*h) / w * sin wt B) mg/k C) x(t) = -mg/k * cos wt + sqrt(2*g*h) / w * sin wt + mg/k

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