Two identical springs, each of force constant k are connected I `(a)` series `(b)` parallel and they support a mass M. Calculate the ratio of the frequency of oscillation of the mass in two systems.

Explanation

(a) For series combination, the effective force constant,
`k_(s)=(kxxk)/(k+k)=(k)/(2)`
`:.` Frequency, `v_(s)=(1)/(2pi)sqrt((k_(s))/(m))=(1)/(2pi)sqrt((k//2)/(m))`
`(1)/(2pi)sqrt((k)/(2m))` …(i)
`(b)` For parallel combination, the effective force constant , `k_(p)=K+k=2k`
`v_(p)=(1)/(2pi)sqrt((k_(p))/(m))=(1)/(2pi)sqrt((2k)/(m))` ...(ii)
`:.` Ration of frequencies , `(v_(s))/(v_(p))=sqrt((k//2)/(2k))=(1)/(2)`