A spring of force constant k is cut in two parts at its one-third lingth. When both the parts are stretched by same amount. The word done in the two parts will be .

Options

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

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Problem
We have a spring of force constant \( k \) that is cut into two parts at its one-third length. We need to determine the work done when both parts are stretched by the same amount.

### Step 2: Determine the Lengths of the Parts
When the spring is cut at one-third of its length:
- The first part (shorter part) has a length of \( \frac{L}{3} \).
- The second part (longer part) has a length of \( \frac{2L}{3} \).

### Step 3: Relate Force Constant to Length
The force constant \( k \) of a spring is inversely proportional to its length:
\[ k \propto \frac{1}{L} \]
Thus, for the two parts:
- For the first part (length \( \frac{L}{3} \)):
\[ k_1 \propto \frac{1}{\frac{L}{3}} = \frac{3}{L} \]
- For the second part (length \( \frac{2L}{3} \)):
\[ k_2 \propto \frac{1}{\frac{2L}{3}} = \frac{3}{2L} \]

### Step 4: Compare the Force Constants
Since \( k_1 \) is proportional to \( 3/L \) and \( k_2 \) is proportional to \( 3/(2L) \), we can see that:
\[ k_1 = \frac{3}{L} > k_2 = \frac{3}{2L} \]
This means that \( k_1 > k_2 \).

### Step 5: Calculate Work Done
The work done on a spring is given by the formula:
\[ W = \frac{1}{2} k x^2 \]
Where \( x \) is the amount of stretch. Since both parts are stretched by the same amount \( x \):
- Work done on the first part:
\[ W_1 = \frac{1}{2} k_1 x^2 \]
- Work done on the second part:
\[ W_2 = \frac{1}{2} k_2 x^2 \]

### Step 6: Compare Work Done
Since \( k_1 > k_2 \):
\[ W_1 > W_2 \]
This indicates that more work is done on the shorter part of the spring.

### Conclusion
The work done is greater for the shorter part of the spring.

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