Question

[2-25 pts) An 41b weight stretches a spring by 24 in. If the weight is pushed up and released 2 ft above the equilibrium with an upward velocity of 8 ft/s. Set up the IVP for this spring mass system. Solve this IVP and rewrite your solution into a single sinusoid. Then, find the amplitude, period, and initial phase of the motion. Find the 3rd time when the weight reaches the highest point. How many times the weight will reach the equilibrium point within the first 5 minutes?

Answer 1

mg 2. 2 m=4&b, X, = 24 in=2ft spring constant of sporing KX =) k = mg= 4x32.174 = 64.348 lb/s? Vu = 8ftls Vmax mg-kx=m mg=kh, =) kY, -kX = maze ma howice rewel pu m ri 2. m dt² Ik (21-x) = max at t=0, x=0 dt² m dx _kX, tk X=0 dt²

now consider that anew variable k

2 dt2 m dt2 X'= X-X, , x = x'+x, d²(X) d² (x²+x1) dx' -k K' a t² solution of this equation will by alt)=Asin (Wt+o) at t=0 x=0 o=Asin (0+0) A&O, sin(wt+0)=0 2 sino=0

So, 0=0,TT, 2T. X(t) = Alinwt) at t=0, v=dx = 8ftle. at

v=dx awcoslwt),4W dt 64.348 4 4.01radle atto, v=8ffls. V=8ft/8= A w Cos(wxo) 8 = AX 4,01 =) A = 1.995ft X(t) = 1.995 sin(W+) A=1.995~2 ft Amplitude (A=2ft)

Ist 2nd 3rd VA period T=2T - 2T 4.01 T= 1.5bbs initial phase o=0 third time when weight reaches the highest point In one time period weight reaches at the highest 1 time for the third highest t=T + Tt I = 1.566 +1.566+ 1.566 4 4 t=3.528. T+T+T/4

n= te In first 5 minutes t = 5x 60=300s total number of round =191.57 T now considering after 191 round time left 300-TX191 -0.894 s.-te now for 19thround Me to to reach this point time taken is T12= 1.566 -0.7834 te> T12 2

so, weight will touch 2 times. the equilibrium point. In one round weight touches equilibrium paint So, total count. N = 191X 2+1 (N=383