First slide

Applications of SHM

Question

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

Moderate

A

2K
B

k
C

k/2
D

2048

Solution

$\frac{1}{k_{eff}} = \frac{1}{k} + \frac{1}{2 k} + \frac{1}{4 k} + \frac{1}{8 k} + . ...$
$= \frac{1}{k} [1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + . ....] = \frac{1}{k} (\frac{1}{1 - 1 / 2}) = \frac{2}{k}$
(By using sum of infinite geometrical progression $a + \frac{a}{r} + \frac{a}{r^{2}} + . .. \infty$ sum (S) $= \frac{a}{1 - r}$ )
$∴ k_{eff} = \frac{k}{2} .$

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