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.

Options

A. `sqrt((K_(a)+K_(b))/(m))`
B. `sqrt((K_(a)K_(b))/((K_(a)+K_(b))m))`
C. `sqrt((K_(a)K_(b))/(4m(K_(a)+K_(b))))`
D. `sqrt((4K_(a)K_(b))/((K_(a)+K_(b))m))`

Answer

C

Explanation

Let `T` be the tension in the cord and `x_(a)` and `x_(b)` the displacements of pulleys `A` and `b` respectively. Now assume that pulley `B` is fixed, then extension of spring `x_(b) = (x)/(2)` or `x = 2x_(b)`. Similarly if we imagine that pulley `A` is fixed, `x = 2x_(a)`.But neither pulley `B` nor pulley `A` is fixed. `:. x = 2x_(a) +2x_(b) ....(1)`
From free body diagram of pulleys,
`2T = k_(b)x_(b) ...(2)` or `2T = k_(a)x_(a) ...(3)`
If `k_(eq)` denotes equivalent spring constant,
`(T)/(k_(eq)) =x =2x_(a) +2x_(b)` From equations (2) and (3)
`x_(a) = (2T)/(k_(a))` and `x_(b) = (2T)/(k_(b)) i.e., k_(eq) = (1)/(4((1)/(k_(a))+(1)/(k_(b))))`
Hence `omega = sqrt((k_(eq))/(m)) = sqrt((k_(a)k_(b))/(4m(k_(a)+k_(b))))`

2nd Method