Question

() attimet woma wo spring system, the displacements of the two masses are represented by (l) and The motion is governed by the following pair of second order simultaneous differential equations az + by zdy d2 where a -20, -3, -100,--25 (a) The system is initially released from rest. Find the partouw solutions to the differential equations which satisfy the initial displacements x For this part of the question give x as a function of t. Omit the "x " (8 marks) 20 andy-0 1 0 This question accepts numbers or formulas Plot Help Switch to Equation Editor Preview (b) What is y as a function of t Omit the "y=" (8 marks) This question accepts numbers or formulas Plot Help Switch to Equation Editor Preview (c) What is the value of xatt-0.13? Give your answer to AT LEAST THREE PLACES OF DECIMALS. Put in the This question accepts numbers or formulas Plot Help Switch to Equation Editor Preview (d) What is the value of y at-0.132 Give your answer to AT LEAST THREE PLACE buvo sure to EST MORE PLACES OR DECIMUS PAR The value on a oma va yra mais This question accepts numbers or formulas Piot] Help Switch to Equation Editor Preview re) What is the period of oscillation in the normal mode corresponding to the larger angular frequency? Give your answer to AT LEAST THREE PLACES OF DECIMALS. Put in this value onlyle. Omt any letters (2 marks) of This question accepts numbers or formulas Piet Help Switch to Equation Editor Preview What is the period of oscillation in the normal mode corresponding to the smaller angular frequency? Give your answer to AT LEAST THREE PLACES OF DECIMALS. Put in this value only e. Omt any letters 2 mans

Answer 1

Ami- briven, = axtby d x = axtby = cx+dy - - - 20% +37 and = -1688–254 - Subjected to initial condition's - x(0)=-20 6 Initial displacement y (0)=-6S À (0)=0 1 Initial velocity (it is zero ý (0) = 0 J as the system is initially at rest) from eqr ① we can write Using eq" ♡ in the eqr ② we get (14.-202--622-25=(-23)

- d* + SO4 + 2sex-80 x =0 w dlare +5 do +4*=0 — equ ② is a lineare. homogenous differential eq". They wring x= A emx we get A (m" + 5 m²+4) e mx=0 - a m7+5m²+4=0 as A emx to for non- - trivial solution * (2) *2.m*. 3 +)+ 4-6)* = 0 a m2 = -1/2 + 3 /4 , - 2/2 - 3/2 = -1,-4 ml, 21 • run the solution is - - X(t) = @ 0:* A GJ X + B finx) +.e 0:* (as 22 + D£in2x) + C 6622 +D Sinza = A Cosa + B fina • Theme - Asink of Blos? Ax Bax -20 Sin 22 + 2D G2X

yle) = }(x-208) I- A loja - B sina - 4 C los 2x - 4D Lin 2x - 20 A loja -20 B sink - 200 C3 22-20D Sin2 = -7 A loja - 7B Link -8C63 2n-8D Sin 2x G(): 14 -2 - 78 x 3 + 16 : đ.ề 2x – 16 D = 2? Applying the [ 0 |-7 o To I 10 -7 initial conditions we get - 1 67TA1 7 -20 -8 ol o 2 0 -16 ^ [M] {7}={F} > {z}=/ ^ 389=[M]='{f} LAI 8 10 07 ~ 42}=16 / دن ہے 0 0 81| 1-7 -1 0 0 Lo 0 - 2 2 2 .

- L Als-1661 Thun c 146 ID 10 Ten final solution is given by [&(t) = -166 lost + 146 cojet I Am @ Ty (4) = 1162 Cog t. – 1168 Cog24Am ☺ © At K = 0.13 x (0.13) = -166 Co3 0.13 + 146 Cos (2.0:13) -20.00 Am @ At t=0.13 y (0.13) = 1162 G3 0.13 – 1168 GJ (2.0-13) =-5.991 Am @ @ larger angulare frequency we=2 Period of oscilliation T, = 2 2 = A Am @ 1. ① smaller angulare frequency Ws=1 Period of oscilliations To = 27 = 20 AM WAS