A spring with a 3-kg mass has a spring constant 48 and neglect friction. The spr

Question

A spring with a 3-kg mass has a spring constant 48 and neglect friction. The spring is stretched 1/2 meters to the left (i.e. negative) and then given an initial velocity of 2 m/sec to the right. Find the position y(t) of the mass after t seconds. Your answer should be a function of the variable t with the general form c_1 e^at cos (beta t) + c_2 e^at sin (beta t) alpha = 0 beta = c_1 = c_2 = Next, use the conversion A = Squareroot c_1^2 + c_2^2 and tan (gamma) = c_1/c_2 (in appropriate quadrant) to rewrite y(t) in the A sin (beta t + gamma) form:

Explanation / Answer

Y = A sin t + B cost,

Y = Atsin t + Bt cost.

Y' = A sin t +At cost + B cost Btsin t

Y " = 2A cost Atsin t 2B sin t Bt cost,

2A cost Atsin t 2B sin t Bt cost + Atsin t + Bt cost = sin t;

2A cost 2B sin t = sin t

We conclude that A = 0 and B = 1/2, so that Y = 1 2 t cost. Answer: y = 1 2 t cost + c1 sin t + c2 cost.

y=c1e

y=ec

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