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 $k\text{'}$. They are connected in parallel and the force constant is $k"$. Then $k\text{'}:k"$is:

Answer 1

Step1: Given data

1. The spring constant of the spring is $k$.
2. The length of the spring is divided into three pieces of ratio $1:2:3$.
3. The spring constant of the system of spring when they are in series is $k\text{'}$.
4. The spring constant of the system of spring when they are in parallel is $k"$ .

Step2: Spring constant

1. The spring constant of a spring is the force exerted on the spring of unit length.
2. Spring constant is defined by the form, $K=\frac{F}{x}$, where, F is the force on the spring due to extension, and x is the amount of extension.
3. So, $springcons\tan t\propto \frac{1}{length}$.

Step3: Diagram

Step4: Finding the spring constants

Let L be the length of the spring and it is cut into three segments of ratio $1:2:3$. So, the three segments are $\frac{L}{6}$, $\frac{L}{3}$, $\frac{L}{2}$.

As we know $springcons\tan t\propto \frac{1}{length}$.So, the spring constant of the spring segments are

$K_1=6K$, $K_2=3K$, $K_3=2K$

So, the spring constant of the spring system when they are connected in series is

$$\begin{gathered} K\text{'}=K_1+K_2+K_3=6K+3K+2K \\ orK\text{'}=11K.......................\left(1\right) \end{gathered}$$

Again, the spring constant of the spring system when connected parallel is

$$\begin{gathered} \frac{1}{K\text{'}\text{'}}=\frac{1}{K_1}+\frac{1}{K_2}+\frac{1}{K_3}=\frac{1}{6K}+\frac{1}{3K}+\frac{1}{2K} \\ or\frac{1}{K\text{'}\text{'}}=\frac{1+2+3}{6K}=\frac{1}{K} \\ or\frac{1}{K\text{'}\text{'}}=\frac{1}{K}.....................\left(2\right) \end{gathered}$$

Step5: Finding the ratio

Now, the ratio of the two given spring systems is

$$\begin{gathered} \frac{K\text{'}}{K\text{'}\text{'}}=\frac{11K}{K}=11\left(From1and2\right) \\ orK\text{'}:K\text{'}\text{'}=11:1 \end{gathered}$$

Therefore, the ratio of spring constant is $K\text{'}:K\text{'}\text{'}=11:1$.