Question

Consider the vibration of mass spring system given by the initial value problem m d²x dt2 dx +b. dt + kx = 0 x(0)=0, x'(0) = 1 Where m, b, k are nonnegative constants and b2 < 4mk. Show that a solution to the problem is given by b2 2m e 2m sin 4mk-b2 4mk 2m t (CO2:P01 - 8 Marks) b. A 200 g mass stretches a spring 5 cm. If it is release from rest from a position 10 cm below the equilibrium position, i. Determine the equation of the motion of the mass ii. Find the frequency of this motion III. Find the period of the motion. (CO2:P01 - 12 Marks)

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Answer 1

Wis mdu itb. dre du ka=0 de dă 28 dn tworo dort ardu dt b ar = m IK m Let a= A é be the Then, ✓ xxarx two x=-281/48% 4wo 2 - of Jarw Now for 6²<4mk b < ymi 8c wo hence we wrili X=-81 i Two-gor then the general solo es giben ihren as 210) -ot A e sim Jurort .8t ot t Be cor Two-8"t) - Now dre It -8.e.iA sintwo-fi- +B CorTwo-y's two uzye [Acollwirt -B Sin Twerf B- da del 'tro of Jail gr A=d A- Two r. then we writé = X(t) &t- .e Sintaro Two-gor IK putting wo Ym v } am Te bt zm get alt) 2m Tymk-br rb.) A mass e sin Tyne-b am 200 g = 0.268 stretch a npring a=5cm 1x 10 m if the spring const is k then mga Kre granitatio Nal alch k = mg Mar 0.249.8 o.of IK = 39.2 Nm mass-0 -Now the est of motion of the -spring nyntem is, m It ka da dt (") Nat Natural fres Wo- K 39.2 rad/s W-14 radla ees Time perioden T- 271 e ans 19 IT=0-44875 К. m n (b) pan-ca) the comes ~ da dtr .-196a subject to inétial cond No=10cm .o.im z'cos