Question

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

Short answer

(b)

Step-by-step solution

Evaluate Statement -1

We are given that the length of the pendulum is increased by 3%. Let's calculate the new length (L_new) and its impact on the periodic time (P_new). L_new = L + 0.03L = 1.03L Now let's calculate the new periodic time (P_new) using the formula P_new = 2π√(L_new/g): P_new = 2π√((1.03L)/g) Next, let's find the percentage change in the periodic time: Percentage change = ( (P_new - P) / P ) × 100 Replacing P_new and P with their expressions: Percentage change = ( ( 2π√((1.03L)/g) - 2π√(L/g) ) / (2π√(L/g)) ) × 100 After simplification: Percentage change ≈ 1.5% Statement -1 is true.

Evaluate Statement -2

The formula for the period of a simple pendulum is: P = 2π√(L/g) From this formula, we can see that the periodic time (P) is proportional to the square root of the length (L) of the pendulum. Hence, Statement -2 is only partially true, as it does not mention the square root relation but only implies a direct proportion.

Identify the correct option

Statement -1 is true. Statement -2 is partially true, but not the correct explanation for Statement -1 because it does not provide the correct relationship between the periodic time and the length of the pendulum. Therefore, the correct option is: (b) Statement -1 is true; statement -2 is true, but statement -2 is not the correct explanation of Statement -1.

Key concepts

Periodic Time

The periodic time of a simple pendulum refers to the time it takes for one complete cycle of its swing from one side to the other and back again. It's an essential aspect of pendulum motion, giving insight into how pendulums operate. To calculate the periodic time, the formula is used:\[P = 2\pi\sqrt{\frac{L}{g}}\]where:

• \(P\) is the periodic time,
• \(L\) is the length of the pendulum,
• \(g\) is the acceleration due to gravity, and
• \(2\pi\) is a constant.

This formula shows that the periodic time does not depend on the mass of the pendulum or the amplitude of the swing, if the amplitude is small enough. This unique property makes the simple pendulum an excellent tool for various scientific and educational purposes.

Length Proportionality

The concept of length proportionality in the context of a simple pendulum describes the relationship between the length of the pendulum and its periodic time. Specifically, the formula for periodic time, \(P = 2\pi\sqrt{L/g}\), indicates that the periodic time is proportional to the square root of the pendulum's length.To put it simply, when the length of the pendulum is increased, the periodic time also increases, but not directly. Instead, it increases proportionally to the square root of the length. This is why statement -2 in the original exercise is partially correct but doesn't capture the full nature of the relationship.To visualize this:

• If the pendulum's length doubles, the periodic time increases by a factor equal to the square root of 2.
• Conversely, if the length decreases, the periodic time decreases, following the same square root proportion.

Understanding this proportionality is crucial for accurate predictions and applications involving pendulums.

Percentage Change

Percentage change measures how much a quantity has changed, expressed as a percentage of its original value. It can be found by comparing the new value against the original in any context, such as the length or periodic time of a pendulum.In the simple pendulum exercise, the length was increased by 3\%, and we were tasked with determining its effect on the periodic time. By expressing the change using the periodic time formula, it was identified that a 3\% increase in length brings about a 1.5\% change in periodic time.Here's how:

• The formula \(P_{\text{new}} = 2\pi\sqrt{\frac{1.03L}{g}}\) provides the new periodic time after the length is increased.
• The percentage change calculation used showed that this change equates to approximately 1.5\%.

Calculating percentage changes helps us understand how modifications to one part of the system will ripple through, impacting other connected variables.

Proportional Relation

A proportional relation in physics describes how one quantity changes relative to another. In a simple pendulum, we see several instances of proportional relationships, such as how the periodic time depends on the square root of the pendulum's length.The formula \(P = 2\pi\sqrt{L/g}\) clearly indicates this relationship, as we've explored. Here are some key learnings about proportional relations from the exercise:

• Proportional relationships help in predicting how a change in one variable, such as length, will affect another, like periodic time.
• Understanding these relations requires acknowledging not just direct proportionality but also factors like square roots that modify the direct relationship.

Grasping these concepts is fundamental for analyzing not just pendulums but any system that involves interconnected variables behaving proportionally.