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 the length of a simple pendulum is increased by $3 \%$, then the periodic time changes by $1.5 \%$. Statement $-2:$ Periodic time of a simple pendulum is proportional to its length. (A) a (B) $b$ (C) $c$ (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 the length of a simple pendulum is increased by $3 \%$, then the periodic time changes by $1.5 \%$. Statement $-2:$ Periodic time of a simple pendulum is proportional to its length.
(A) a
(B) $b$
(C) $c$
(D) d

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Key Concepts

Simple Pendulum

A simple pendulum refers to an idealized pendulum consisting of a mass suspended by a light, inextensible string in a uniform gravitational field. Its behavior is often used to study periodic motion, and its dynamics are described by simple harmonic motion under small oscillation approximations.

Pendulum Period Formula

The period of a simple pendulum is given by the formula T = 2??(L/g), where T is the time period, L is the length of the pendulum, and g is the acceleration due to gravity. This relationship illustrates that the period is proportional to the square root of the length, not directly proportional to the length, which is key when evaluating changes in the period with respect to changes in length.

Proportionality and Percentage Change

Understanding proportional relationships helps in assessing how changes in one variable affect another. In the context of a pendulum, because the period is proportional to the square root of the length, a percentage change in length leads to a smaller percentage change in the period. This concept is essential when comparing statements about changes in physical quantities.

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