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. $ \frac{k}{2} $
• D. $ 2048 $

Answer 1

Answer: (C)

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,..$
$\therefore \frac{1}{k_{e}}=\frac{1}{k}+\frac{1}{2 k}+\frac{1}{4 k}+\frac{1}{8 k}+..$
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.
$\Rightarrow \frac{1}{k_{e}}=\frac{1}{k} \times \frac{1}{\left(1-\frac{1}{2}\right)}=\frac{2}{k}$
$\Rightarrow k_{e}=\frac{k}{2}$