Infinite springs with force constants k, 2k, 4k and 8k... respectively are connected in series. The effective force constant of the spring will be

Question

Infinite springs with force constants k, 2k, 4k and 8k... respectively are connected in series. The effective force constant of the spring will be (A) 2k  (B) k (C)  (k/2)  (D)  2048

Tardigrade Admin · Accepted Answer

Effective force constant is equal to the reciprocal of the sum of individual force constants, hence

\frac{1}{k _{e}} = \frac{1}{k _{1}} + \frac{1}{k _{2}} + \frac{1}{k _{2}} + ...

Given,

k_{1} = k . k_{2} = 2 k, k_{3} = 4 k, ..

∴ \frac{1}{k _{e}} = \frac{1}{k} + \frac{1}{2 k} + \frac{1}{4 k} + \frac{1}{8 k} + ..

The given series is a geometric progression series, hence sum is

S_{\infty} = \frac{a}{1 - r}

where,

a

is first term of series and

r

the common difference.

\Rightarrow \frac{1}{k _{e}} = \frac{1}{k} \times \frac{1}{( 1 - \frac{1}{2} )} = \frac{2}{k}

\Rightarrow k_{e} = \frac{k}{2}

Q. Infinite springs with force constants $k, 2 k, 4 k$ and $8 k ...$ respectively are connected in series. The effective force constant of the spring will be

$2 k$

$k$

$\frac{k}{2}$

$2048$

Q. Infinite springs with force constants k,2k,4k and 8k... respectively are connected in series. The effective force constant of the spring will be

2k

k

2k​

2048

Q. Infinite springs with force constants $k, 2 k, 4 k$ and $8 k ...$ respectively are connected in series. The effective force constant of the spring will be

$2 k$

$k$

$\frac{k}{2}$

$2048$