A spring of force constant `K` is cut in two parts at its one third length. When both the parts are stretched by same amount, the work done in the two parts, will be `:-`

Options

A. equal in both
B. greater for the longer part
C. greater for the shorter part
D. data insufficient.

Explanation

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the Spring Constants
We have a spring of force constant \( K \) that is cut into two parts at one-third of its length. Let's denote the total length of the spring as \( L \). After cutting, we have two parts:
- Part 1 (shorter part): Length = \( \frac{L}{3} \)
- Part 2 (longer part): Length = \( \frac{2L}{3} \)

When a spring is cut, the spring constant of each part changes. The spring constant \( k \) of a spring is inversely proportional to its length. Therefore, if we denote the spring constants of the two parts as \( k_1 \) and \( k_2 \):
- For Part 1: \( k_1 = 3K \) (since it is shorter)
- For Part 2: \( k_2 = \frac{3K}{2} \) (since it is longer)

### Step 2: Calculate the Work Done on Each Part
The work done on a spring when it is stretched by an amount \( x \) is given by the formula:
\[ W = \frac{1}{2} k x^2 \]
We will calculate the work done for both parts when they are stretched by the same amount \( x \).

- For Part 1:
\[ W_1 = \frac{1}{2} k_1 x^2 = \frac{1}{2} (3K) x^2 = \frac{3K x^2}{2} \]

- For Part 2:
\[ W_2 = \frac{1}{2} k_2 x^2 = \frac{1}{2} \left(\frac{3K}{2}\right) x^2 = \frac{3K x^2}{4} \]

### Step 3: Compare the Work Done
Now we can compare the work done on both parts:
- Work done on Part 1: \( W_1 = \frac{3K x^2}{2} \)
- Work done on Part 2: \( W_2 = \frac{3K x^2}{4} \)

### Step 4: Conclusion
From the calculations, we can see that:
\[ W_1 > W_2 \]
This means that more work is done on the shorter part of the spring compared to the longer part when both are stretched by the same amount.

### Final Answer
The work done in the two parts will be greater for the shorter part.