Question

A spring of force constant $k$ is cut into lengths of ratio $1:2:3$. They are connected in series and the new force constant is $k'$. Then they are connected in parallel and force constant is $k''$ the $k':k''$ is?
(A). $1:14$
(B). $1:6$
(C). $1:9$
(D). $1:11$

Answer 1

The Hooke’s law gives us the relation between force, spring constant and the displacement of the spring. Using the relation we can calculate new lengths and hence new force constants. When springs are connected in parallel, the equivalent of force constant is the sum of inverse of its constants. When the spring pieces are connected in series, the equivalent is the sum of constants.

Formulas used:
$F=-kx$
$\dfrac{1}{k'}=\dfrac{1}{{{k}_{1}}}+\dfrac{1}{{{k}_{2}}}+\dfrac{1}{{{k}_{3}}}$
$k''={{k}_{1}}+{{k}_{2}}+{{k}_{3}}$

Complete step-by-step solution:
Force constant of a spring is the force acting per unit length.
$F=-kx$
Here,
$F$ is the force acting on spring
$k$ is the force constant
$x$ is the displacement of spring.

From the above equation,
$k\propto \dfrac{1}{x}$

If the new lengths are $\dfrac{x}{6},\,\dfrac{x}{3},\,\dfrac{x}{2}$ their force constants will be $6k,\,3k,\,2k$respectively.

When the spring constant are connected in series, then the equivalent of their spring constants will be-
$\dfrac{1}{k'}=\dfrac{1}{{{k}_{1}}}+\dfrac{1}{{{k}_{2}}}+\dfrac{1}{{{k}_{3}}}$
$\begin{aligned} & \Rightarrow \dfrac{1}{k'}=\dfrac{1}{6k}+\dfrac{1}{3k}+\dfrac{1}{2k} \\\ & \Rightarrow \dfrac{1}{k'}=\dfrac{6}{6k} \\\ \end{aligned}$
$\therefore k'=k$ - (1)

When the spring pieces are connected in parallel, the equivalent of their spring constants will be-
$k''={{k}_{1}}+{{k}_{2}}+{{k}_{3}}$
$\Rightarrow k''=6k+2k+3k$
$\therefore k''=11k$ - (2)

From eq (1) and eq (2), the ratio of $k':k''$ is-
$\begin{aligned} & \dfrac{k'}{k''}=\dfrac{k}{11k} \\\ & \Rightarrow \dfrac{k'}{k''}=\dfrac{1}{11} \\\ \end{aligned}$
$\therefore k':k''=1:11$

The ratio $k':k''$ is $1:11$, therefore, the correct option is (D).

Additional Information:
Motions which repeat itself after equal intervals of time are called periodic motions. The number of periods taking place in unit time is called frequency. Time period is the time taken to complete one period of motion. Harmonic motion is a special case of periodic motion.

Note:
The motion of the spring is periodic and harmonic motion. The principle of harmonic motion is that the force acting on the object is proportional to the negative of displacement. It oscillates about a mean position periodically. A restoring force develops in the string towards its equilibrium position.