A body of mass `m` hangs from a smooth fixed pulley `P_(1)` by the inextensible string fitted with the springs of stiffness `k_(1)` and `k_(2)` . The string passes over the smooth light pulley `P_(2)` which is connected with another ideal spring of stiffness `k_(2)`. Find the period of oscillation of the body.

Answer

`2 pi m ((1)/(k_(3)) + (4)/(k_(2)) + (1)/(k_(1)))`

Explanation

Let `x_(1), x_(2) and x_(3)` be the elongations of the spring `k_(1). K_(2) and k_(3)` repectively. The total displacement of m is
`x = x_(1)+ 2 x_(2) + x_(3)`
put `x_(1) = (mg)/(k_(1)). x_(1) = (2 mg)/(k_(2))and x_(3) = (mg)/(k_(3))`
to obtain `x = mg [(1)/(k_(3)) + (4)/(k_(2)) + (1)/(k_(1))]`
Then, `T = 2 pi sqrt((x)/(g)) = 2 pi m ((1)/(k_(3)) + (4)/(k_(2)) + (1)/(k_(1)))`