Question

QUESTION 2 (20 MARKS) a Consider the vibration of mass spring system given by the initial value problem dx dx de+b + kx = 0 dt *(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 X(t) = 2m Amk- em sin 4mk-02 2m (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, Determine the equation of the motion of the mass Find the frequency of this motion ili. Find the period of the motion. i. ii. (CO2:P01 - 12 Marks)

Answer 1

(0) 423 + box + x = 0 ; Xlo) = 0; X16)1 Okm char Ean of = mbit botk=0 D = -b = √624 (m)CK) am) As bunk a D = -bt JumK-62 2 (m) X4) = explain t) Carca ( June 2) + Brom A=0 4) Xt = explain tye eine tem) e explanat) or (Sumok-:) exp edint) a Juoks? ?) x(0)=1 ) 0+ (Stmk B) W = 1 ;)B= Junk- runk b? explain the Answell X0 = 0) am Numk-ba am Jumk-ba B 2m t ) ) x't) = 2) am am So X(t) = 2m Gi (Surak-63 Umk-b9 20 t )

اف Spring constant K = Gm chare ea. m= 200 g = m = 0.9 kg ģL= 5 cm = 0.05 m (0.2) (9.8) 39a 0.05 m=0.2 mxll+KX=0 = 0.2x + 39, 2X CO xll +19 6x=0 m+196=0 matlle XGH - A Cos (lut) + B Em clut) X()=0; x(0) - 10 ģ X (0) 20 7 A207 X) = B&m (lut) X'CH IUB OSCIUH ;x60-10 3 148-10 9 8-9 So XU = 5 Grillet) c) frequency = J J K 1 39.2 1 (14) = 7 शा 0.2 ATT - frequency = 1 and Answer Period 7 Period = frequency Period = 1 1 1 Answer 7/21