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. `(1)/(T^(2))=(1)/(T_(1)^(2))+(1)/(T_(2)^(2))`
D. `(1)/(T)=(1)/(T_(1))+(1)/(T_(2))`

Answer

B

Explanation

For the two springs,
`T_(1)=2pi sqrt((m)/(K_(1)), T_(2)=2pi sqrt((m)/(K_(2))`
`therefore T_(1)^(2)+T_(2)^(2)=4pi^(2)m((1)/(K_(1))+(1)/(K_(2)))`
`T_(1)^(2)+T_(2)^(2)=(4pi^(2)m(K_(1)+K_(2)))/(K_(1)+K_(2))`
When the springs are joined in series, the effective force constant is given by `(1)/(K)=(1)/(K_(1))+(1)/(K_(2))=(K_(1)+K_(2))/(K_(1)K_(2)) " or " K=(K_(1)K_(2))/(K_(1)+K_(2))` and for the combined spring
`T=2pi sqrt((m)/(K))=2pi sqrt((m(K_(1)+K_(2)))/(K_(1)K_(2)))`
`therefore T^(2)=(4pi^(2)m(K_(1)+K_(2)))/(K_(1)K_(2))`
from (1) and (2) `T^(2)=T_(1)^(2)+T_(2)^(2)`