Given below are two statements:
Statement I A second's pendulum has a time period of 1 s.
Statement

Q: Why is it called a 'second's' pendulum if its time period is 2 seconds?

It is called a second's pendulum because each individual 'tick' or 'tock'—which corresponds to a swing from one extreme to the other (a half oscillation)—takes exactly 1 second.

Q: Does the mass of the bob affect the time period of the pendulum?

No, the time period of a simple pendulum is given by $T = 2\pi\sqrt{\frac{l}{g}}$. It depends only on the length of the string $l$ and the acceleration due to gravity $g$, completely independent of the bob's mass.

Q: What is the approximate length of a second's pendulum on Earth?

Using the formula $T = 2\pi\sqrt{\frac{l}{g}}$, if we plug in $T = 2\text{ s}$ and $g \approx 9.8\text{ m/s}^2$, we find that the length $l$ is approximately $0.994\text{ m}$, which is very close to $1\text{ meter}$.

Q: What is the difference between an oscillation and a beat?

An oscillation is a complete round trip (e.g., Extreme A to Extreme B and back to Extreme A). A beat is just a half oscillation (e.g., Extreme A to Extreme B).

Q: Is the velocity of the pendulum bob constant between the extreme positions?

No, the velocity is zero at the extreme positions where it momentarily stops to reverse direction, and it reaches its maximum velocity as it passes through the lowest point (the mean position).

Question

Given below are two statements:
Statement I A second's pendulum has a time period of 1 s.
Statement II It takes precisely one second to move between the two extreme positions.
In the light of the above statements, choose the correct answer from the options given below.

Options:

(A) Both Statement I and Statement II are false.

(B) Statement I is false but Statement II is true.

(C) Statement I is true but Statement II is false.

(D) Both Statement I and Statement II are true.

Sigma Percentile · Accepted Answer

### The Magic of the Second's Pendulum

Imagine you are looking at an old, majestic grandfather clock. The rhythmic *tick-tock* you hear is governed by a very special physical system known as a **second's pendulum**. This problem tests our fundamental understanding of how this specific pendulum operates.

### Analyzing Statement I

Let's start with the definition. What exactly makes a pendulum a "second's" pendulum? By definition, a second's pendulum is one whose time period is exactly $2	ext{ s}$.

The time period $T$ is the time taken to complete one full oscillation. A full oscillation means the bob starts from one extreme position, swings all the way to the opposite extreme, and then returns to its original starting point.

Statement I claims that the time period of a second's pendulum is $1	ext{ s}$. Since we know the true time period is $T = 2	ext{ s}$, Statement I is definitively **false**.

### Analyzing Statement II

Now, let's break down the motion of the pendulum. If a full round trip (Extreme 1 $ightarrow$ Extreme 2 $ightarrow$ Extreme 1) takes $2	ext{ s}$, how long does it take to complete just half of that journey?

The time taken to move from one extreme position to the other extreme position is exactly half of the total time period. Mathematically, we can write this as:

$$t = \frac{T}{2}$$

Substituting our known time period $T = 2	ext{ s}$:

$$t = \frac{2}{2} = 1	ext{ s}$$

This means every single swing across the center takes precisely one second. This is exactly what Statement II claims! Therefore, Statement II is **true**.

### Conclusion

Combining our findings, we can confidently conclude that Statement I is false, but Statement II is true. This perfectly matches option (b).

Select Answer:

Visualized Solution

$The Second’s Pendulum$

$Time Period Definition$

$Evaluating Statement I$

$Analyzing the Motion$

$Evaluating Statement II$

$Final Conclusion$

The Sigma Insight: Simple Harmonic Motion (SHM)

The Magic of the Second's Pendulum

Analyzing Statement I

Analyzing Statement II

Conclusion

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The Second’s Pendulum

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Time Period Definition

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluating Statement I

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Analyzing the Motion

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Evaluating Statement II

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Conclusion