A particle connected at the end of a spring executes SHM with a period `T_(1)`. While the corresponding period for another spring is `T_(2)`. When the two springs are connected inseries and particle is ocnnected at the one T. Find the relation between T, `T_(1)` and `T_(2)`

Explanation

Leat a force F be applied to the series combination of springs and `x_(1)` and `x_(2)` be the displacement in the two springs. Then
`F=k_(1)x_(1)=-k_(2)x_(2)or x_(1)=-(F)/(k_(1))` and `x_(2)=-(F)/(k_(2))`
Total extension, `x=x_(1)+x_(2)`
`=F[(1)/(k_(1))+(1)/(k_(2))]=-F((k_(1)+k_(2))/(k_(1)k_(2)))`
or `F=((k_(1)k_(2))/(k_(1)+k_(2)))x`
Comparing it with the force equation `F=-kx` We have, force constant of the series combination of springs
`k=((k_(1)k_(2))/(k_(1)+k_(2)))`
Here, `T_(1)=2pisqrt((m)/(k_(1))),T_(2)=2pisqrt((m)/(k_(2)))`
`T=2pisqrt((m)/(k))=2pisqrt((m(k_(1)+k_(2)))/(k_(1)k_(2)))=2ppisqrt((m)/(k_(1))+(m)/(k_(2)))`
or `T^(2)=4pir^(2)((m)/(k_(1))+(m)/(k_(2)))=(2pisqrt((m)/(k_(1))))^(2)+(2pisqrt(m)/(k_(2)))^(2)`
or `T^(2)=T_(1)^(2)+T_(2)^(2)`