When a body is suspended from two light springs separately, the periods of vertical oscillations are `T_1` and `T_2`. When the same body is suspended from the two spring connected in series, the period will be

Options

A. `T_(1)`
B. `T_(2)`
C. `T_(1)+T_(2)`
D. `sqrt(T_(1)^(2)+T_(2)^(2))`

Answer

D

Explanation

`T_(1)^(2)=(4pi^(2)m)/(k_(1))andT_(2)^(2)=(4pi^(2)m)/(k_(2))`
when the springs are connected in series the spring constant are
`K_(s)=(k_(1)k_(2))/(k_(1)+k_(2))" As "K_(s)prop(1)/(T_(s)^(2))`
`therefore (1)/(T_(s)^(2))=((1)/(T_(1)^(2))xx(1)/(T_(2)^(2)))/((1)/(T_(1)^(2))+(1)/(T_(2)^(2)))=((1)/(T_(1)^(2))xx(1)/(T_(2)^(2)))/((T_(2)^(2)+T_(1)^(2))/(T_(1)^(2)xxT_(2)^(2)))`
`(1)/(T_(s)^(2))=(1)/(T_(1)^(2)+T_(2)^(2)) therefore T_(s)=sqrt(T_(1)^(2)+T_(2)^(2))`