A particle at the end of a spring executes simple harmonic motion with a period `t_(1)` while the corresponding period for another spring is `t_(2)` if the oscillation with the two springs in series is T then

Options

A. `T=t_(1)+t_(2)`
B. `T^(2)=t_(1)^(2)+t_(2)^(2)`
C. `T^(1)=t_(1)^(-1)+t_(2)^(-1)`
D. `T^(2)=t_(1)^(-2)+t_(2)^(-2)`

Explanation

`t_(1)=2pisqrt((m)/(k_(1)))" " ....(1),`
`t_(2)=2pisqrt((m)/(k_(2)))" " ....(2),`
when springs are in series then
`T=2pisqrt(((m)/(k_(1)k_(2)))/(k_(1)+k_(2)))=2pisqrt(m(k_(1)+k_(2)))/(k_(1)k_(2))`
squaring and adding (1) and (2) we get
`t_(1)^(2)+t_(1)^(2)=4pi^(2)(m)/k_(1)+4pi^(2)(m)/k_(2)`
`=4pi^(2)m(k_(1)+k_(2)/(k_(1)k_(2)))`
or `t_(1)^(2)+t_(2)^(2)=T^(2)`