For the following questions, statement as well as the...

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement $-1$ is true, statement $-2$ is true; statement $-2$ is the correct explanation of statement $-1$. (b) Statement $-1$ is true, statement $-2$ is true but statement $-2$ is not the correct explanation of statement $-1$. (c) Statement $-1$ is true, statement $-2$ is false (d) Statement $-1$ is false, statement $-2$ is true (A) a (B) $b$ (C) $\mathrm{c}$ (D) $\mathrm{d}$ Statement $-1$ : If a spring having spring constant $\mathrm{k}$ is divided into equal parts, then the spring constant of each part will be $2 \mathrm{k}$. Statement $-2:$ When the length of the elastic spring is increased ( stretched) by $\mathrm{x}$, then the amount of work required to be done is $(1 / 2) \mathrm{kx}^{2}$ (A) a (B) $\mathrm{b}$ (C) $\mathrm{c}$ (D) d

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option.
(a) Statement $-1$ is true, statement $-2$ is true; statement $-2$ is the correct explanation of statement $-1$.
(b) Statement $-1$ is true, statement $-2$ is true but statement $-2$ is not the correct explanation of statement $-1$.
(c) Statement $-1$ is true, statement $-2$ is false
(d) Statement $-1$ is false, statement $-2$ is true
(A) a
(B) $b$
(C) $\mathrm{c}$
(D) $\mathrm{d}$
Statement $-1$ : If a spring having spring constant $\mathrm{k}$ is divided into equal parts, then the spring constant of each part will be $2 \mathrm{k}$. Statement $-2:$ When the length of the elastic spring is increased ( stretched) by $\mathrm{x}$, then the amount of work required to be done is $(1 / 2) \mathrm{kx}^{2}$
(A) a
(B) $\mathrm{b}$
(C) $\mathrm{c}$
(D) d

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Key Concepts

Hooke's Law

Hooke's Law is a fundamental principle in mechanics which states that the force required to displace a spring from its equilibrium position is directly proportional to the displacement, expressed as F = kx. This linear relationship underpins the behavior of elastic systems and serves as the basis for further analysis of spring dynamics.

Elastic Potential Energy

Elastic potential energy refers to the energy stored in a spring when it is deformed (stretched or compressed) from its equilibrium position. The energy is given by the expression (1/2)kx², where k is the spring constant and x is the amount of displacement. This relationship describes the work done on the spring and is key in energy conservation problems involving elastic materials.

Dependence of Spring Constant on Length

For a uniform spring, the spring constant is inversely proportional to its length. When a spring is divided into equal parts, each segment behaves like a spring with a spring constant adjusted according to its reduced length. This concept is crucial for understanding how alterations in a spring’s geometry affect its mechanical properties, such as in series combinations where shorter springs have higher stiffness constants.

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