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'. Then k' : k" is :

Options

A. `1:6`
B. `1:9`
C. `1:11`
D. `1:14`

Answer

To solve the problem step by step, we will analyze the situation of cutting the spring into three parts, connecting them in series, and then in parallel.

### Step 1: Determine the lengths of the springs
We are given that the spring is cut into lengths in the ratio of 1:2:3. Let's denote the total length of the spring as \(6L\) (this is just a convenient total length that maintains the ratio). Therefore, the lengths of the individual springs will be:
- First spring: \(L_1 = L\)
- Second spring: \(L_2 = 2L\)
- Third spring: \(L_3 = 3L\)

### Step 2: Calculate the effective spring constants for each part
The spring constant \(k\) is inversely proportional to the length of the spring. Thus, we can calculate the new spring constants for each part:
- For the first spring of length \(L\):
\[ k_1 = \frac{k \cdot 6L}{L} = 6k \]
- For the second spring of length \(2L\):
\[ k_2 = \frac{k \cdot 6L}{2L} = 3k \]
- For the third spring of length \(3L\):
\[ k_3 = \frac{k \cdot 6L}{3L} = 2k \]

### Step 3: Calculate the equivalent spring constant when connected in series
When springs are connected in series, the equivalent spring constant \(k'\) can be calculated using the formula:
\[ \frac{1}{k'} = \frac{1}{k_1} + \frac{1}{k_2} + \frac{1}{k_3} \]
Substituting the values:
\[ \frac{1}{k'} = \frac{1}{6k} + \frac{1}{3k} + \frac{1}{2k} \]
Finding a common denominator (which is \(6k\)):
\[ \frac{1}{k'} = \frac{1}{6k} + \frac{2}{6k} + \frac{3}{6k} = \frac{6}{6k} = \frac{1}{k} \]
Thus, we find:
\[ k' = k \]

### Step 4: Calculate the equivalent spring constant when connected in parallel
When connected in parallel, the equivalent spring constant \(k''\) is simply the sum of the individual spring constants:
\[ k'' = k_1 + k_2 + k_3 \]
Substituting the values:
\[ k'' = 6k + 3k + 2k = 11k \]

### Step 5: Find the ratio of the two spring constants
Now we need to find the ratio \(k' : k''\):
\[ \frac{k'}{k''} = \frac{k}{11k} = \frac{1}{11} \]

### Final Answer
Thus, the ratio \(k' : k''\) is:
\[ \boxed{\frac{1}{11}} \]