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_(p)ltT_(1)ltT_(2)ltT_(s)`
B. `T_(s)^(2)=T_(1)^(2)+T_(2)^(2)`
C. `sqrt(T_(s))=sqrt(T_(1))+sqrt(T_(2))`
D. `(1)/(T_(p)^(2))=(1)/(T_(1)^(2))+(1)/(T_(2)^(2))`

Answer

A, B, D

Explanation

`T_(1)=2pisqrt((m)/(k_(1))), T_(2)=2pisqrt((m)/(k_(2)))`
`T_(s)=2pisqrt((m)/((k_(1)k_(2))//(k_(1)=k_(2))))`
`T_(p)=2pisqrt((m)/((k_(1)+k_(2))))`
From above we note that `T_(s)gtT_(2)gtt_(1)gtT_(p),i.e.,` option (a) is correct.
`(1)/(T_(p)^(2))=(k_(1)+k_(2))/(4pi^(2)m)=(k_(1))/(4pi^(2)m)+(k_(2))/(4pi^(2)m)`
`=(1)/(T_(1)^(2))+(1)/(T_(2)^(2)),i.e.,` option (d) is correct.
`T_(s)^(2)=(4pi^(2)m(k_(1)+k_(2)))/(k_(1)k_(2))=(4pi^(2)m)/(k_(2))+(4pi^(2)m)/(k_(1))`
`=T_(2)^(2)+T_(1)^(2), i.e.,` option (b) is correct.