Question

4.(12 pts. Choose to do either part (a) or part (b). (a) A woman bails out an airplane at an altitude of 10,000 ft with a parachute. Assume linear air resistance force is 1.2mv (where v is the velocity and m is the mass of the woman and the parachute), how long will it take her to reach the ground? (Use gravitational constant g - 32 ft/s). () A mass-spring system with the initial conditions x(0)=0, x'(O) = 2. The motion is an undamped forced motion with mass m=2, the spring constant k = 10 and the external force Fg = 2 sin(51). Find the position function x(i) of the mass-spring system. Is the motion a resonance or a practical resonance or neither?

Answer 1

soln(b) mass spring system undamped torze - m.x" tba't k.x = Fext undamped b=o m=2, k=10, Fext = asinst with initial condition alo)=0, x'(0)=2 - 2x" to.x+10x=asinst =) *"+sx=sinist). Aureliany k²+5=0 ,k=tids - Equation ' a(t)=a cosast+(asin 15t for Monhomogeneous Pent xolt) = Asinst tВ losst xelt)=5A cosst-5B sinst zölt=-25 A sinst -25B cosst - 25.A sinst -25 Brost tstsinst +5 Blosst - sinst Equating coefficient of sin st, wosst. -20A = 1 , Ardo , B=0 General solution - X(t)= 4 cos(ist) + G sin(ist) - sinist) t=0, x10) = 0 Geo x'(t)=-15a sin 15t) +256, coslast)-I cosest) x' 10) = 2.= 0+1562- B C2 = 1 solution a(t) = g_sinlist) - 2 sin(st) 4.15 Resonance frequency w=J K = 25. bet we are not getting above equation ccmofa15 and 5 not possible. We have to say it is practical resonance

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