A block of mass `m` is attached to one end of a light inextensible string passing over a smooth light pulley `B` and under another smooth light pulley `A` as shown in the figure. The other end of the string is fixed to a ceiling. `A` and `B` are held by spring of spring constants `k_(1)` and `k_(2)`. Find angular frequency of small oscillations of small oscillations of the system.

Answer

A, B, D

Explanation

Let `F` be the restoting force (extra tension) on block `m` when displaced by `x` from its equilibrium position.
`x = 2x_(1) + 2x_(2)`
`= 2[(2F)/(k_(1)) + (2F)/(k_(2))]`
`= 4F((k_(1) + k_(2))/(k_(1)k_(2)))`
or `F = - (k_(1)k_(2))/(4(k_(1) +k_(2)))x`
`:. a = - (k_(1)k_(2))/(4m(k_(1) + k_(2)))x`
`omega = sqrt((k_(1)k_(2))/(4m(k_(1) + k_(2))))`