The figure shows an arrangement of mass m with three springs of different springs constant. When m is slightly displaced and then released, find the frequency of its motion.

Explanation

Here, two springs of force constans `k_(2) and k_(3)` are in series and the third spring of force constant `k_(1)` is in parallel with their series combination. Therefore, equivalent force constant of the arrangement is
`k. = (k_(2)k_(3))/(k_(2) + k_(3)) + k_(1)`
The time period is given by
`T = 2pi sqrt((m)/(k.)) = 2pi sqrt((m)/((k_(2)k_(3))/(k_(2)+k_(3))+k_(1)))`
or `T = 2pisqrt((m(k_(2) + k_(3)))/(k_(2)k_(3) + k_(3)k_(1) + k_(3)k_(2)))`
Frequency, `v = (1)/(T)`
`v = (1)/(2pi)sqrt((k_(2)k_(3) + k_(3)k_(1) + k_(1)k_(2))/(m(k_(2) + k_(3))))`