A spring (spring constant=k) is cuttend into 4 equal parts and two parts are connected in parallel. What is the effective spring constant?

Options

A. 4 k
B. 16 k
C. 8 k
D. 6 k

Explanation

To solve the problem of finding the effective spring constant when a spring of spring constant \( k \) is cut into 4 equal parts and two parts are connected in parallel, we can follow these steps:

### Step 1: Understand the relationship between spring constant and length
The spring constant \( k \) of a spring is inversely proportional to its length. This means that if the length of the spring is reduced, the spring constant increases. The relationship can be expressed as:
\[ k \cdot l = \text{constant} \]
where \( l \) is the length of the spring.

### Step 2: Determine the spring constant of each cut part
Since the spring is cut into 4 equal parts, the length of each part will be \( \frac{l}{4} \). Let’s denote the spring constant of each of these smaller springs as \( k_0 \).

Using the relationship from Step 1:
\[ k \cdot l = k_0 \cdot \frac{l}{4} \]
From this, we can solve for \( k_0 \):
\[ k_0 = 4k \]
This means that each of the smaller springs has a spring constant of \( 4k \).

### Step 3: Connect two parts in parallel
When two springs are connected in parallel, the effective spring constant \( k_{\text{effective}} \) is simply the sum of their individual spring constants. Therefore, if we connect two of the smaller springs (each with spring constant \( 4k \)) in parallel, we have:
\[ k_{\text{effective}} = k_0 + k_0 = 4k + 4k = 8k \]

### Conclusion
The effective spring constant when two parts of the cut spring are connected in parallel is:
\[ \boxed{8k} \]

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