Question

A simple spring has length $l$ and force constant $k$. It is cut into two springs of length ${l_1}$ and ${l_2}$ such that ${l_1} = n{l_2}$( n is an integer) the force constant of a spring of length ${l_1}$ is
A. $k(1 + n)$
B. $\dfrac{{nk}}{{n + 1}}$
C. $\dfrac{{\left( {1 + n} \right)k}}{n}$
D. $k$

Answer 1

We know that the length of the spring and force constant are inversely related. Total length $l$ of the spring is given as a sum of lengths ${l_1}$ and ${l_2}$. By substituting for each length in terms of respective force constants and on solving we can find the value of the spring constant of length ${l_1}$ in terms of the total spring constant $k$.

Complete step by step answer:
It is given that a spring has a length of $l$ .
The force constant of the spring is $k$.
Then the spring is cut into two springs of length ${l_1}$ and ${l_2}$ .
Relationship between ${l_1}$ and ${l_2}$ is given as
${l_1} = n{l_2}$
Where, n is an integer.
We need to find the force constant of the spring of length ${l_1}$ .
We know that the length of a spring and force constant are inversely related.
So, we can write it as
$l \propto \dfrac{1}{k}$
$\Rightarrow l = \dfrac{c}{k}$ (1)
Where, c is a constant of proportionality.
Let ${k_1}$ be the force constant of spring with length ${l_1}$ and ${k_2}$ be the force constant of spring with length ${l_2}$.
Then we can write
$\Rightarrow {l_1} = \dfrac{c}{{{k_1}}}$ (2)
And
$\Rightarrow {l_2} = \dfrac{c}{{{k_2}}}$ (3)
We know that the total length $l$ is the sum of ${l_1}$ and ${l_2}$ .
$\Rightarrow l = {l_1} + {l_2}$
$\Rightarrow l = {l_1} + \dfrac{{{l_1}}}{n}$
$\because {l_1} = n{l_2}$
Substituting the value of $l$ and ${l_1}$from equation 1 and 2, we get
$\Rightarrow \dfrac{c}{k} = \dfrac{c}{{{k_1}}} + \dfrac{c}{{n{k_1}}}$
Now let us solve for ${k_1}$ .
$\Rightarrow \dfrac{1}{k} = \dfrac{1}{{{k_1}}} + \dfrac{1}{{n{k_1}}}$
$\Rightarrow \dfrac{1}{k} = \dfrac{{n + 1}}{{n{k_1}}}$
$\Rightarrow {k_1} = \dfrac{{n + 1}}{n}k$
This is the value of the spring constant of length ${l_1}$ .
Hence, the correct answer is option C.

Note: Remember that the length and spring constant are inversely related. If we increase the length of the spring then the force constant will decrease and if you decrease the length of spring then the force constant will increase. So, if we cut a spring in half the spring constant of each half will be doubled. In our case since ${l_1} = n{l_2}$ The relation between spring constants of these parts will then be ${k_2} = n{k_1}$