Infinite springs with force constants k,2k,4k | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) For springs connected in series,the effective force constant $k_{eff}$ is given by the formula: $\frac{1}{k_{eff}} = \frac{1}{k_1} + \frac{1}{k_2}

Vedclass · Accepted Answer

$\frac{k}{2}$

Vedclass · Answer

(C) For springs connected in series,the effective force constant $k_{eff}$ is given by the formula: $\frac{1}{k_{eff}} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} + \dots$Substituting the given values: $\frac{1}{k_{eff}} = \frac{1}{k} + \frac{1}{2k} + \frac{1}{4k} + \frac{1}{8k} + \dots$Factor out $\frac{1}{k}$: $\frac{1}{k_{eff}} = \frac{1}{k} \left( 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots ight)$The term in the bracket is an infinite geometric progression with first term $a = 1$ and common ratio $r = \frac{1}{2}$.The sum $S$ of an infinite geometric progression is given by $S = \frac{a}{1 - r}$.$S = \frac{1}{1 - 1/2} = \frac{1}{1/2} = 2$.Therefore,$\frac{1}{k_{eff}} = \frac{1}{k} 	imes 2 = \frac{2}{k}$.Thus,$k_{eff} = \frac{k}{2}$.

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

Similar Questions

When a weight is applied to a spring,it stretches by an amount $x$. What is the energy stored in it? ($T$ is the tension force developed in the spring and $k$ is the spring constant.)

$A$ massless spring gets elongated by an amount $x_1$ under a tension of $5 \ N$. Its elongation is $x_2$ under a tension of $7 \ N$. For an elongation of $(5x_1 - 2x_2)$,the tension in the spring will be: (in $N$)

Find the acceleration of the $3 \text{ kg}$ mass when the acceleration of the $2 \text{ kg}$ mass is $2 \text{ ms}^{-2}$ as shown in the figure. (in $\text{ ms}^{-2}$)

$A$ force of $20\,dyne$ applied to the end of a spring increases its length by $1\,mm$. What will be the force constant of the spring?

Vedclass Test Series

Exam Paper Generator

Online Exam Module