2. A spring-mass system for an object of mass 2 kg is described by the IVP given

Question

2. A spring-mass

system

for an

object of mass 2 kg is described by the IVP given below,

where the units for

t are seconds and the units

for

y are meters.

2y"+8y'+ 80y = 20 cos(!t); y(0) =

0;y' (0) = 0

a. Take

w= 0:5.

i. Find the steady-state solution

and

write it in the form

Rcos(wt-p)

ii. Give the general solution of

the

ODE.

iii. Find the solution to the IVP by applying the initial

conditions.

b. Using the formula for

wmax from Section 4.6,

determine

the value of ! for which

the amplitude

R of the steady-state

solution

would be maximized.

c. Repeat i-iv for

w= wmax.

Explanation / Answer

L = T - V = 1/2 (I/R^2 + m) v^2 + mg x - 1/2 k x^2 for which, the Lagrange equation is (see http://en.wikipedia.org/wiki/Hamiltonian… (I/R^2 + m) d^2 x /dt^2 + k x = mg which is the regular expression for the harmonic oscillation with frequency sqrt(k/(m+M/2))

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