Infinite springs with force constants `k, 2k, 4k, 8k,…..` respectively are connected in series to which mass 'm' is attched. The frequency of small oscillations is……..

Options

A. `f = (1)/(2pi) sqrt((2K)/(m))`
B. `f = (1)/(2pi) sqrt((K)/(m))`
C. `f = (1)/(2pi) sqrt((K)/(2m))`
D. `f= (1)/(2pi) sqrt((2048 K)/(m))`

Answer

C

Explanation

`(1)/(k_(eff))=(1)/(k)+(1)/(2k)+(1)/(4k)+(1)/(8k)+..........`
`= (1)/(k) [1+(1)/(2)+(1)/(4)+(1)/(4)+(1)/(8)+......]`
`S_(oo) = (a)/(1-r) = (1)/(1-(1)/(2))=2`
`(1)/(k_(eff))=(1)/(k)xx 2 rArr k_(eff) = (k)/(2)`
`f = (1)/(2pi) sqrt((k)/(2m))`