Question

A spring of force constant k is cut into two pieces such that one piece is double
the length of the other. Then the long piece will have a force constant of
A. (2/3) k
B. (3/2) k
C. 3k
D. 6k

Answer 1

Hint:Recall the expression for the spring force and determine the proportionality relation between force constant and elongation in the spring. Express the force constant for the longer piece and substitute the elongation in the longer piece and force constant of the spring. The force constant of the longer piece should be in terms of the force constant of the original spring.

Formula used:
Restoring force, \[{F_s} = kx\]
Here, k is the force constant and x is the elongation of the spring.

Complete step by step answer:
We have the expression for the spring force,
\[{F_s} = kx\]
\[ \Rightarrow k = \dfrac{{{F_s}}}{x}\]
Here, k is the force constant and x is the elongation in the spring.
Therefore, we can write the expression,
\[k \propto \dfrac{1}{x}\]
\[ \Rightarrow k = \dfrac{c}{x}\] …… (1)
Here, c is the constant.
Let \[{x_1}\] be the length of the first piece and therefore the length of the second piece will
be\[2{x_1}\]. Therefore, the total length of the spring will be,
\[{x_1} + 2{x_1} = x\]
\[ \Rightarrow {x_1} = \dfrac{x}{3}\] …… (2)
Let’s express the force constant for the longer piece as follows,
\[{k_2} = \dfrac{c}{{2{x_1}}}\]
Using equation (1) and (2) in the above equation, we get,
\[{k_2} = \dfrac{{kx}}{{2\left( {x/3} \right)}}\]
\[ \Rightarrow {k_2} = \dfrac{3}{2}k\]
Therefore, the force constant for the longer piece will be (3/2) k.

So, the correct answer is option (B).

Note:The unit of force constant is N/m and it measures how strong the restoring force the spring is. The spring force is inversely proportional to the elongation of the spring. Note that the spring force is proportional to the elongation of the spring and not the total length of the spring.