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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 spring A is given by

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 spring A is given by

(a) 5k/2

(b) 3k/5

(d) k

Solution :-

For a spring, the spring constant is inversely proportional to its length, this is because the spring force is given by :-

$\displaystyle F=-kx$

$\displaystyle \Rightarrow k(in\text{ }magnitude)=\frac{F}{x}$

$\displaystyle \Rightarrow k\propto \frac{1}{x}$

Here x is the extension/compression in the spring.

Now the extension/compression is directly proportional to the length of the spring, i.e., x ∝ l

$\displaystyle \Rightarrow k\propto \frac{1}{x}\propto \frac{1}{l}$

Now, in this question let l be the original length of the spring, then

$\displaystyle k\propto \frac{1}{l}$ …..(1)

And as

$\displaystyle {{l}_{A}}:{{l}_{B}}=2:3$

$\displaystyle \Rightarrow {{l}_{A}}=\frac{2}{5}l\text{ }\!\!\And\!\!\text{ }$

$\displaystyle {{l}_{B}}=\frac{3}{5}l$

$\displaystyle \Rightarrow {{k}_{A}}\propto \frac{1}{{{l}_{A}}}\propto \frac{5}{2l}$ …..(2)

From equations (1) and (2), we get

$\displaystyle \frac{{{k}_{A}}}{k}=\frac{5}{2l}\times \frac{l}{1}$

$\displaystyle \Rightarrow {{k}_{A}}=\frac{5}{2}k$

Hence option (a) is correct.

If a spring of stiffness k is cut into two parts A and B of length lA : lB = 2 : 3, then the stiffness of spring A is given by