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) $\dfrac{{2K}}{3}$
B) $\dfrac{{3K}}{2}$
C) $3K$
D) $6K$

Answer 1

Hint: We know that the ideal spring obeys Hooke's law which defines the force on the spring is $F = Kx$ . The spring of force constant K is given in the question and we know that the force constant is inversely proportional to the length of the spring. So, if the length of the spring is more, then it will have a smaller value of force constant and if the length of the spring is less, then it will have a greater value of force constant. From the question, we are having two pieces of spring so we can calculate the length of each piece and then, the length can be used to find the force constant of the long piece.

Formula used:
The force constant is inversely proportional to the length of the spring which can be mathematically written as:
$
  K \propto \dfrac{1}{l} \\
  K = \dfrac{C}{l} \\
 $
Where,
$K = $ Force constant of the spring
$C = $ Proportionality constant
$l = $ Length of the spring

Complete step by step solution:
According to the question, the force constant of the spring is K.
Now, the spring is cut into two pieces such that the length of one piece is double the length of the other piece.
Let us assume the spring is cut into the lengths of $x$ and $2x$ .
So, the sum of the two lengths will be equated to $l$ .
$
  x + 2x = l \\
  3x = l \\
  x = \dfrac{l}{3} \\
 $
We know that the force constant can be mathematically written as
 $
  K = \dfrac{C}{l} \\
  C = Kl...\left( 1 \right) \\
 $
So, we can write the force constant for the long piece which is:
${K'} = \dfrac{C}{{2x}}$
Now, using the value of C from equation (1)
$
  {K'} = \dfrac{C}{{2x}} \\
  {K'} = \dfrac{{Kl}}{{2x}} \\
 $
Now, putting the value of $x$ in the above equation, we get
$
  {K'} = \dfrac{{Kl}}{{2 \times \dfrac{l}{3}}} \\
  {K'} = \dfrac{{3K}}{2} \\
 $
Hence, the correct answer is option B.

Note: As we know that the force constant of spring is inversely proportional to the length of the spring. So, we deduce an expression with proportionality constant to get the formula in terms of the given data. Then, the lengths of the respective pieces can be put in the expression to get the force constant of the spring.