In the arrangement shown in figure, pulleys are light and spring are ideal. `K_(1)`, `k_(2)`, `k_(3)`and `k_(4)` are force constant of the spring. Calculate period of small vertical oscillations of block of mass `m`.

Answer

A, B, C, D

Explanation

When the mass `m` is displaced from its mean position by a distance `x`, let `F` be the restoring (extra tension) force produced in the string. By this extra tension further elongation in the springs are
`(2F)/(k_(1))` , `(2F)/(k_(2))`, `(2F)/(k_(3))` and `(2F)/(k_(4))` respectively.
Then,
`x = 2((2F)/(k_(2))) + 2((2F)/(k_(2))) + 2((2F)/(k_(3))) + 2((2F)/(k_(4)))`
or `F((4)/(k_(1)) + (4)/(k_(2)) + (4)/(k_(3)) + (4)/(k_(4))) = - x`
Here negative sign shows the restoring nature of force.
`a = - (x)/(m((4)/(k_(1)) + (4)/(k_(2)) + (4)/(k_(3)) + (4)/(k_(4)))`
`T = 2pi sqrt |(x)/(a)|`
`= 4pi sqrt (m((1)/(k_(1)) + (1)/(k_(2)) + (1)/(k_(3)) + (1)/(k_(4))))`