Find the values of k such that the massspring system describ

Find the value(s) of k, such that the mass-spring system described by each of the equations below is undergoing resonance. 2u\" + ku = 9 sin 5t 7u\" + ku = pi cos t

Solution

a)

Equatoin can be written as

u\'\'+k/2 u=9/2 sin(5t)

Natural frequency is sqrt{k/2} assuming k is positive

So for resonanc we need

sqrt{k/2}=5

k=50

b)

Equation can be written as

u\'\'+k/7 u=pi /7 cos(t)

Natural frequency =sqrt{k/7} which should be equal to forcing frequency

Hence,

sqrt{k/7}=1

k=7