Springs of spring constants k, 2k, 4k, 8k, …… are connected in series. A mass m kg is attached to the lower end of the last spring and the systemm is allowed to vibrate. What is the time period of oscillations? Given `m=40gm,k=2.0Ncm^(-1)`.

Explanation

Here, `m=40g=0.40kg,`
`k=2.0Ncm^(-1)=2.0xx100Nm^(-1)`
Effective spring constant k is given by
`(1)/(K)=(1)/(k)+(1)/(2k)+(1)/(4k)+(1)/(8k)+......`
`=(1)/(k)[(1)+(1)/(2)+(1)/(4)+(1)/(8)+....]`
`=(1)/(k)[(1)/(1-1//2)]=(2)/(k) or K=k//2`
`:.` Time period, `T=2pisqrt((m)/(K))`
`=2ppisqrt((m)/(k//2))=2pisqrt((2m)/(k))`
`=2xx(22)/(7)xxsqrt((2xx0.04)/(2.0xx100))`
`=0.126s`