Question

I dt2 7. Use the second order solver method (the symbolic one) to solve the two following ODES dy dy (a) + + 16y = 0 dt day dy (b) + 16 = 0 dt2 dt both with initial conditions y(0) = 10, y'(0) = 0 and plot their solutions. These ODEs can model the position of a pendulum swinging through a fluid, where one equation represents the pendulum being in molasses, and the other is in water. Looking at both the graphs and the equations, which is which? Explain your reasoning. + 10

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Answer 1

The sinesoidal graph of molasses

704) -1 + -63 2 2 y colato yll + Ýtity (D+D+6)y=0 mt m +16=0 -1+ Ni-4646) br-uac = -63co this will be under damping (Complex rooks) model m: -{ + Vezi -45°ß x = -1/2, B = √6% y(t) = { [C, cospt a co simple Cam Baba & Sund y(): 7 °C G Coscold Cz8inco) [ac] =) (6=10] g'o)=0 = y't)= ett { - Gues sin Eest) + (6 563 √63 Cossos (c, cassezt 4+2 sin van het (3) y'ro) =ě ° (-660)+C2CJES (1) J -260[G (1) +6] [ Cases ] - Ža C2 563 = 29 С. lo 40 J63 T (2=10 J63 2 (10) x 2 4.63 10.cos ſatt J63 éš [ in molasses. YCH ) sin Test which is solution of pendulum swinging The exponential facti ett has a negative exponent and therefore givis the decaying amplitude As too, the exponental goes asymptotically too so .yees goes asymptotically to its equilibrium pository y=0 also

10 5 a -15 -10 -5 5 10 15

75) Y" +10y'+16y =o (117+160+16) y = 0 m: -107 V100-64 mrtiom +16 ایک پین - - 9 0 2 2 my+ 8m+2m+16 AS mcm+8) + 2(M+8)=0 bruaryo (m+2)(m+8) = 0 thes will be over damping mi:-2, me = -8 model Y(A) = q ezt + C2 est gro)= a ě "+(2 é o y ros=10 =) po - -81- y! (A) = 29 e 21 -8 cz yko) - 2G -SC2 -26 - S2 =) G+4C2=0 2 G + C2 =10 G+4C2 = 0 -362 =10 --) (22-10/3 G = 10-62 = 10tro 3 40 라 -st 3 C2=-113 (9=40 e 4284 which is Thi walú when the damping solution of pendulum swinging ui is large the fictional force is great that the system can not oscillate it might sound odd, but an unforced overdamped harmonic oscillatos does not oscillate Since both exponents negalue every solutions in this Case goes asympótically to the equilibrium a=0 an
The graph of pendulum swinging in water

10- 5. -15 -10 -5 5 10 15 -5 --0 -