Question

A spring of spring constant 'k' cut into n pieces. What is the spring constant of each piece?

• A. $\frac{k}{n}$
• B. $\frac{n}{k}$
• C. $\frac{n^2}{k}$
• D. $nk$

Hint:

Intuitively, making a spring shorter drastically reduces the amount of elastic material available to stretch, rendering it much stiffer and significantly harder to pull!

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The effective stiffness or spring constant $k$ of a uniform coiled spring depends heavily on its total unstretched natural length.
Specifically, the spring constant is entirely inversely proportional to the original length of the spring material.
Step 2: Key Formula or Approach:
The defining mathematical relationship is $k \propto \frac{1}{L}$, which means the product of the spring constant and its length is always a constant value: $k_{\text{initial}} L_{\text{initial}} = k_{\text{final}} L_{\text{final}}$.
Step 3: Detailed Explanation:
Let the initial spring constant strictly be $k$ and the initial total length be $L$.
When this uniform spring is systematically cut into $n$ perfectly equal smaller pieces, the new length of each individual piece naturally becomes $L' = \frac{L}{n}$.
Using our established inverse relationship $k \times L = \text{constant}$, we can directly equate the initial and final states:
\[ k \times L = k' \times L' \] Substitute the new reduced length into the equation:
\[ k \times L = k' \times \left( \frac{L}{n} \right) \] We safely cancel out the common length variable $L$ from both sides of the equation:
\[ k = \frac{k'}{n} \] Rearranging to formally solve for the new spring constant $k'$:
\[ k' = nk \] Step 4: Final Answer:
The newly determined spring constant of each individual piece is genuinely $nk$.