Infinite springs with force constant k , 2k , 4k and 8k .... respectively are connected in series. e

English

Gujarati

Hindi

13.Oscillations

medium

Infinite springs with force constant $k$, $2k$, $4k$ 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

$\frac{k}{4}$

Solution

(c)$\frac{1}{{{k_{eff}}}} = \frac{1}{k} + \frac{1}{{2\,k}} + \frac{1}{{4\,k}} + \frac{1}{{8\,k}} + ….$
$ = \frac{1}{k}\left[ {1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + …..} \right]$$ = \frac{1}{k}\left( {\frac{1}{{1 – 1/2}}} \right)$$ = \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}}$)
$\therefore {k_{eff}} = \frac{k}{2}.$

Standard 11

Physics

Share

Similar Questions

A mass m is suspended from a spring of length l and force constant $K$. The frequency of vibration of the mass is ${f_1}$. The spring is cut into two equal parts and the same mass is suspended from one of the parts. The new frequency of vibration of mass is ${f_2}$. Which of the following relations between the frequencies is correct

easy

A mass of $5\, {kg}$ is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. A simple pendulum of length $4\, {m}$ has the same period of oscillation as the spring system. What is the value of acceleration due to gravity on the planet where these experiments are performed? (In ${m} / {s}^{2}$)

hard

(JEE MAIN-2021)

Two springs with negligible masses and force constant of $K_1 = 200\, Nm^{-1}$ and $K_2 = 160\, Nm^{-1}$ are attached to the block of mass $m = 10\, kg$ as shown in the figure. Initially the block is at rest, at the equilibrium position in which both springs are neither stretched nor compressed. At time $t = 0,$ a sharp impulse of $50\, Ns$ is given to the block with a hammer.

hard

If a spring has time period $T$, and is cut into $n$ equal parts, then the time period of each part will be

easy

(AIEEE-2002)

A block of mass $m$ is suspended separately by two different springs have time period $t_1$ and $t_2$ . If same mass is connected to parallel combination of both springs, then its time period will be

hard