Question

The force constants of two springs are \[{K_1}\]and \[{K_2}\]. Both are stretched till their elastic energies are equal. If the stretching forces are \[{F_1}\] and \[{F_2}\] then \[{F_1}:{F_2}\]is,
A. \[{K_1}:{K_2}\]
B. \[{K_2}:{K_1}\]
C. \[\sqrt {{K_1}} :\sqrt {{K_2}} \]
D. \[K_1^2:K_2^2\]

Answer 1

Hint:The elastic potential energy of the spring is proportional to the square of the change in length of the spring and the spring force is linear proportional to the change in the length. As the spring forces are different, so to have same elastic potential energy, the change in lengths of springs will be different.

Formula used:
\[U = \dfrac{1}{2}k{x^2}\]
Where U is the elastic potential energy in the spring of spring constant K and x is the change in length.
\[F = kx\]
Where F is the magnitude of the spring force of spring constant K when change in length is x.

Complete step by step solution:
Let the changes in length of the springs are \[{x_1}\]and \[{x_2}\]. Then the elastic energy in first spring is,
\[{U_1} = \dfrac{1}{2}{K_1}x_1^2\]

The elastic energy in the second spring is,
\[{U_2} = \dfrac{1}{2}{K_2}x_2^2\]

It is given that the elastic energy stored in both the springs are equal.
\[{U_1} = {U_2}\]
\[\dfrac{1}{2}{K_1}x_1^2 = \dfrac{1}{2}{K_2}x_2^2\]
\[\dfrac{{{x_1}}}{{{x_2}}} = \sqrt {\dfrac{{{K_2}}}{{{K_1}}}} \]
The spring forces in the springs are given as \[{F_1}\]and \[{F_2}\]

Using force formula, the forces in the springs are,
\[{F_1} = {K_1}{x_1}\]for the first spring,
\[{F_2} = {K_2}{x_2}\]for the second spring.

On dividing the expression for the forces, we get
\[\dfrac{{{F_1}}}{{{F_2}}} = \dfrac{{{K_1}{x_1}}}{{{K_2}{x_2}}}\]
\[\dfrac{{{F_1}}}{{{F_2}}} = \dfrac{{{K_1}}}{{{K_2}}} \times \left( {\sqrt {\dfrac{{{K_2}}}{{{K_1}}}} } \right)\]
\[\dfrac{{{F_1}}}{{{F_2}}} = \sqrt {\dfrac{{{K_1}}}{{{K_2}}}} \]
So, the spring force ratio is \[{F_1}:{F_2} = \sqrt {{K_1}} :\sqrt {{K_2}} \]

Therefore, the correct option is (C).

Note:The work done by the stretching force is stored as the elastic potential energy of the spring. For the same amount of applied force, the change in length of the springs will be inversely proportional to the spring constant.