Question

If a spring of stiffness $k$ is cut into two parts $A$ and $B$ of length ${l_A} = {l_B} = 2:3$, then the stiffness of the spring $A$ is given by:
(A) $\dfrac{{3k}}{5}$
(B) $\dfrac{{2k}}{5}$
(C) $k$
(D) $\dfrac{{5k}}{2}$

Answer 1

The stiffness of the spring can be determined by using the Hooke’s law. This law gives the relation between the stiffness of the spring and the length of the spring. Given information is only the length ratio. By using this law, the stiffness can be determined.

Formula used:
By Hooke’s law, the relation between the stiffness of the spring and the length of the spring is given by,
$k \propto \dfrac{1}{L}$
Where $k$ is the stiffness of the spring and $L$ is the length of the spring.

Complete step by step answer:
Given that,
The length ratio of the spring is, ${L_A}:{L_B} = 2:3$
If the length of the spring is $L$, then, ${L_A} = \dfrac{{2L}}{5}$
If the length of the spring is $L$, then, ${L_B} = \dfrac{{3L}}{5}$
By Hooke’s law, the relation between the stiffness of the spring and the length of the spring is given by,
$k \propto \dfrac{1}{L}\,...............\left( 1 \right)$
If the initial spring constant is $k$ then, then from equation (1),
$\Rightarrow kL = {k_A}{L_A} = {k_B}{L_B}$
Now,
$\Rightarrow kL = {k_A}{L_A}$
The stiffness of the spring $A$ is given by,
$\Rightarrow {k_A} = \dfrac{{kL}}{{{L_A}}}$
By substituting the value of ${L_A}$ in the above equation, then the above equation is written as,
$\Rightarrow {k_A} = \dfrac{{kL}}{{\left( {\dfrac{{2L}}{5}} \right)}}$
By rearranging the above equation, then the above equation is written as,
$\Rightarrow {k_A} = \dfrac{{5 \times kL}}{{2L}}$
By cancelling the same term $L$ in the above equation, then the above equation is written as,
$\Rightarrow {k_A} = \dfrac{{5k}}{2}$
Thus, the above equation shows the stiffness of the spring $A$.

Hence, the option (D) is the correct answer.

Note:
By using this same procedure, the stiffness of the spring $B$ is also determined. But here the stiffness of the spring $A$ is asked, so the stiffness of the spring $A$ is determined. By equation (1), the stiffness is inversely proportional to the length of the spring.