Question

An ideal, massless spring with spring constant 1 N/m (upper panel of the given figure) is cut into 5 equal parts. If two of these parts are connected in parallel (lower panel of the given figure), what is the resultant spring constant in N/m? (rounded off to the nearest integer)
\includegraphics[width=0.5\linewidth]{59.png}

Hint:

When springs are connected in parallel, the resultant spring constant is the sum of the individual spring constants. Also, when a spring is cut into \( n \) parts, the spring constant of each part increases by a factor of \( n \).

Answer 1

Given:
The spring constant of the original spring is \( k = 1 \, {N/m} \),
The spring is cut into 5 equal parts.
When a spring is cut into \( n \) equal parts, the spring constant of each part increases by a factor of \( n \). Therefore, the spring constant of each of the smaller springs is: \[ k_{{part}} = n \cdot k = 5 \cdot 1 = 5 \, {N/m} \] Now, two of these parts are connected in parallel. When springs are connected in parallel, the resultant spring constant \( k_{{total}} \) is the sum of the spring constants of the individual parts: \[ k_{{total}} = k_{{part}} + k_{{part}} = 5 \, {N/m} + 5 \, {N/m} = 10 \, {N/m} \] Thus, the resultant spring constant when two parts are connected in parallel is: \[ k_{{total}} = 10 \, {N/m} \]