Suppose a spring-mass system is modelled by the following IVP. u\" + 5u = 0 u(0)

Question

Suppose a spring-mass system is modelled by the following IVP.
u" + 5u = 0
u(0) = 15
u'(0) = -30
Suppose the mass has a size of 4 kilograms, where u is the displacement of the
spring from gravitational equilibrium, in meters.
(a) What is the spring constant k of the spring? What are its units?
(b) Describe the state of the spring-mass system at time t = 0, in words.
(c) Solve the system for u(t).
(d) Express u(t) in the form Rcos(mu of t - delta), and find the natural frequency,
frequency, period, amplitude, and phase of the mass-spring system.
(e) Sketch a graph of y = u(t), over 3 periods of the motion.

Explanation / Answer

u" + 5u = 0
u(0) = 15
u'(0) = -30

characterstic equation given by

x^2+5 = 0

x = +i5 or -5i

solution given by

U(x) = C1 cos5x + C2 sin5x

U(0) = 15 = C1

U'(x) = -5 C1 sin5x + 5 C2 cos5x

U'(0) = -30 = 5 C2

C2 = -30 / 5

U(x) = 15 cos5x -30 / 5 sin5x

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