If the length of second’s pendulum is increased by 2%, how many seconds will it lose per day?

1. Basics of Periodic and Oscillatory Motion (basic)

In the study of mechanics, we begin with the most fundamental way nature keeps time: Periodic Motion. Any motion that repeats itself at regular intervals of time is called periodic motion. Imagine the rotation of the Earth or the hands of a clock; they return to the same position after a fixed duration. As noted in Science-Class VII, Measurement of Time and Motion, p.109, this predictability is what allows us to measure time effectively.

A specific and very important type of periodic motion is Oscillatory Motion. This occurs when an object moves "to and fro" (back and forth) about a central point, known as the mean position. While all oscillatory motions are periodic because they repeat their path in fixed intervals, not all periodic motions are oscillatory (for instance, the Earth’s orbit is periodic but not a back-and-forth oscillation). The most classic example is the simple pendulum, which consists of a small metallic ball called a bob suspended by a thread Science-Class VII, Measurement of Time and Motion, p.109.

When you displace the bob and release it, it begins to oscillate. The time it takes to complete one full "to-and-fro" movement—starting from one point and returning to it after swinging to the other side—is defined as the Time Period. Interestingly, the time period of a simple pendulum of a fixed length is remarkably constant at any given place Science-Class VII, Measurement of Time and Motion, p.118. This stability is driven by forces like gravity, a non-contact force that constantly pulls the bob back toward its mean position Science, Class VIII, Exploring Forces, p.77.

Feature	Periodic Motion	Oscillatory Motion
Definition	Repeats after a fixed time interval.	Repeats "to and fro" about a mean position.
Example	Earth revolving around the Sun.	A child on a swing or a pendulum.
Relationship	The broader category.	A specific type of periodic motion.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118; Science ,Class VIII . NCERT(Revised ed 2025), Exploring Forces, p.77

2. The Simple Pendulum and its Time Period (basic)

To understand how we measure time, we must first look at the simple pendulum. A simple pendulum consists of a small metallic ball, called a bob, suspended from a rigid stand by a thin, non-stretchable string. When you pull the bob slightly to one side and release it, it begins to swing back and forth. This is a classic example of periodic or oscillatory motion. One complete oscillation occurs when the bob moves from its center (mean) position to one extreme, then to the other extreme, and finally back to the center Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109. The time taken to complete this one full cycle is defined as the Time Period (T) Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.110.

The most critical discovery about the pendulum is that its time period is constant at a given location for a fixed length Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118. This regularity is what made the pendulum the heart of mechanical clocks for centuries. The relationship is governed by the formula: T = 2π√(L/g), where L is the length of the string and g is the acceleration due to gravity. Notice that the mass of the bob does not appear in this formula; experiments show that changing the bob's mass or material has no effect on how long a swing takes Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.110.

Because the time period is proportional to the square root of the length (T ∝ √L), even small changes in length can affect the clock's accuracy. A very useful approximation for small variations is that the percentage change in the time period is roughly half the percentage change in length (ΔT/T ≈ ½ * ΔL/L). For instance, if a string expands and becomes 2% longer, the time period increases by about 1%, meaning the pendulum swings slower and the clock 'loses' time.

Factor	Does it affect the Time Period?	Relationship
Length (L)	Yes	Time increases as length increases (T ∝ √L)
Gravity (g)	Yes	Time decreases as gravity increases (T ∝ 1/√g)
Mass of Bob	No	No effect on the period
Amplitude (small swing)	No	The swing stays constant regardless of small width changes

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.110; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118

3. Defining the Second’s Pendulum (basic)

A Second’s Pendulum is a special type of simple pendulum designed specifically to measure time in a very intuitive way. While the time period of any pendulum is the time it takes to complete one full oscillation—moving from a starting point, reaching both extremes, and returning to the start Science-Class VII, Measurement of Time and Motion, p.109—a second’s pendulum is defined by having a time period of exactly 2 seconds.

You might wonder why it is called a "second's" pendulum if the period is two seconds. This is because each single swing (moving from one extreme position to the opposite extreme) takes exactly 1 second. This property made it the fundamental component of early precision clocks, like the famous grandfather clocks, where every "tick" or "tock" represents exactly one second passing. Because the time period depends on the length of the string and the local gravity Science-Class VII, Measurement of Time and Motion, p.110, the length of a second’s pendulum on Earth is always approximately 0.994 meters (roughly 1 meter). If you were to take this same pendulum to the Moon, where gravity is weaker, it would swing more slowly and would no longer have a 2-second period.

In a standard 24-hour day, which contains 86,400 seconds, a second's pendulum will complete exactly 43,200 full oscillations. This mathematical consistency is why the pendulum’s motion is described as periodic—it repeats its path in fixed, predictable intervals Science-Class VII, Measurement of Time and Motion, p.109. If the length of the pendulum changes even slightly (perhaps due to metal expansion in summer), the time period will shift away from 2 seconds, causing the clock to either "lose" or "gain" time relative to the standard day.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109; Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.110

4. Variations in Gravity (g) and its Impact (intermediate)

To understand how a pendulum clock behaves in different parts of the world, we must first recognize that acceleration due to gravity (g) is not a universal constant. The Earth is not a perfect sphere but an oblate spheroid, bulging at the equator and flattened at the poles. Consequently, the distance from the Earth's center to its surface is greater at the equator than at the poles. Since gravity follows an inverse-square law with distance, gravity is stronger at the poles and weaker at the equator FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19. Additionally, the uneven distribution of mass within the Earth's crust creates 'gravity anomalies,' meaning the local value of g can vary based on the density of the material beneath your feet.

This variation has a direct impact on the time period (T) of a simple pendulum, which is governed by the formula T = 2π√(L/g). In this relationship, the time period is inversely proportional to the square root of gravity. If you take a pendulum clock from the equator to the North Pole, the value of g increases. As g increases, the time period T decreases, meaning the pendulum swings faster and the clock 'gains' time. Conversely, moving toward the equator (where g is lower) causes the time period to increase, making each swing take longer and causing the clock to 'lose' time or run slow.

Engineers and physicists also have to account for changes in the length (L) of the pendulum, often caused by thermal expansion. For small variations, we use a rule of thumb: the fractional change in the time period is approximately half the fractional change in length (ΔT/T ≈ ½ * ΔL/L). A standard second's pendulum has a period of 2 seconds (one second for each 'tick'). Since there are 86,400 seconds in a day, even a tiny 1% increase in the time period (caused by a 2% increase in length) would cause the clock to lag by 864 seconds in a single day (0.01 * 86,400). Understanding these sensitivities is crucial for precision instrumentation in different geographical contexts.

Location	Gravity (g)	Pendulum Speed	Clock Status
Poles	Higher	Faster Swings (Lower T)	Gains Time
Equator	Lower	Slower Swings (Higher T)	Loses Time

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.19; Exploring Society: India and Beyond. Social Science-Class VI . NCERT (Revised ed 2025), Locating Places on the Earth, p.14

5. Resonance and Forced Oscillations (intermediate)

To understand Resonance, we must first recognize that every object has a natural frequency—a specific rate at which it "prefers" to vibrate if disturbed. For example, a simple pendulum has a natural time period (and thus a frequency) determined primarily by its length and gravity Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109. If you nudge a pendulum and let it go, it will swing at this natural frequency. However, in the real world, objects are often subjected to an external, rhythmic force. This leads us to the concept of forced oscillations.

A forced oscillation occurs when an external periodic force acts on a system, compelling it to vibrate not at its own natural frequency, but at the frequency of the applied force (the driving frequency). Imagine pushing a child on a swing. If you push at random intervals, the swing moves awkwardly. But if you synchronize your pushes with the swing's natural rhythm, the height of the swing increases dramatically. This special state, where the driving frequency matches the system's natural frequency, is called Resonance. At resonance, the system absorbs the maximum possible energy from the external source, leading to a massive increase in the amplitude (the scale) of the vibration.

Feature	Free/Natural Oscillation	Forced Oscillation	Resonance
Frequency	System's natural frequency	Frequency of the external driver	Driver frequency = Natural frequency
Amplitude	Constant (in ideal vacuum)	Small/Moderate	Reaches a maximum peak
Energy Transfer	None (internal energy)	Low efficiency	Highly efficient energy transfer

We see resonance across all branches of science. In telecommunications, for instance, high-frequency radio waves interact with free electrons in the ionosphere. When the wave frequency is suitable, it causes these electrons to vibrate and re-radiate energy back to Earth Physical Geography by PMF IAS, Earths Atmosphere, p.279. Conversely, resonance can be destructive. During an earthquake, if the frequency of the ground's shaking matches the natural frequency of a building, the structure will experience violent swaying, often leading to severe damage or collapse, even if the earthquake's energy isn't exceptionally high Physical Geography by PMF IAS, Earthquakes, p.182.

Sources: Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109; Physical Geography by PMF IAS, Earths Atmosphere, p.279; Physical Geography by PMF IAS, Earthquakes, p.182

6. Errors, Approximations, and Fractional Changes (intermediate)

In physics and engineering, we rarely deal with perfect numbers. Whether it is measuring the focal length of a lens or the oscillation of a pendulum, small errors or changes are inevitable. To handle these, we use Approximations. A fundamental rule in error analysis is that for a quantity calculated using a power relationship (like y = xⁿ), the fractional change in the result is approximately equal to the power multiplied by the fractional change in the variable. This is a powerful shortcut that allows us to bypass complex calculations when changes are small (typically less than 10%).

Consider the Simple Pendulum, where the time period (T) is given by the formula T = 2π√(L/g). In this relationship, T is proportional to the square root of length, or L^1/2. By applying our approximation rule, the fractional change in the time period (ΔT/T) is roughly 1/2 the fractional change in length (ΔL/L). This means if the length of a pendulum increases by 2% due to thermal expansion, the time period does not increase by 2%, but by approximately 1%. This logic of relating variables is as universal as the lens and mirror formulas used to calculate distances in optics Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.143, 155.

Understanding these changes is vital for timekeeping. A Seconds Pendulum is designed to have a period of exactly 2 seconds (taking 1 second for each swing). There are 86,400 seconds in a standard day. If the period increases (the pendulum takes longer to swing), the clock is said to be "slow" and it loses time. Conversely, if the period decreases, the clock runs fast and "gains time." For example, a 1% increase in the time period translates to the clock losing 1% of the total seconds in a day (0.01 × 86,400 = 864 seconds). Much like delays in regulatory clearances can affect the efficiency of an industry Indian Economy, Vivek Singh (7th ed. 2023-24), Supply Chain and Food Processing Industry, p.366, these small fractional changes in mechanics compound into significant real-world errors over time.

Variable Change	Effect on Period (T)	Clock Behavior
Length (L) Increases	T Increases (Slower)	Loses Time
Length (L) Decreases	T Decreases (Faster)	Gains Time

Sources: Science, class X (NCERT 2025 ed.), Light – Reflection and Refraction, p.143, 155; Indian Economy, Vivek Singh (7th ed. 2023-24), Supply Chain and Food Processing Industry, p.366

7. Timekeeping and Daily Loss/Gain in Clocks (exam-level)

To understand how clocks lose or gain time, we must first look at the foundation of our timekeeping: the mean solar day. This is the average 24-hour period it takes for the Sun to return to its highest point in the sky, a cycle governed by the Earth's rotation Science, Class VIII, Keeping Time with the Skies, p.178. In mechanical clocks, we use a pendulum to subdivide this day into precise seconds. As discovered by Galileo and later refined by Christiaan Huygens, the time taken for a pendulum to complete one oscillation (its time period, T) is remarkably constant for a given length at a specific location Science, Class VII, Measurement of Time and Motion, p.108.

The physics of this motion is defined by the formula T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. Crucially, the time period depends only on these two factors and is independent of the mass of the bob Science, Class VII, Measurement of Time and Motion, p.110. If either L or g changes, the clock will no longer accurately reflect a standard second. For example, on a hot day, a metal pendulum rod might expand due to heat. When the length (L) increases, the time period (T) also increases. Because each 'tick' now takes slightly longer than it should, the clock falls behind the actual time — we say the clock "loses time" or is "running slow." Conversely, if the rod contracts in winter, the period shortens, and the clock "gains time" or "runs fast."

When dealing with small changes, we use a handy approximation: the fractional change in the time period is roughly half the fractional change in length (ΔT/T ≈ ½ * ΔL/L). To calculate the total daily loss or gain, we multiply this fractional change by the total seconds in a day (86,400 seconds). For instance, if a 2% increase in length leads to a 1% increase in the time period, the clock would lose 1% of 86,400 seconds, resulting in a loss of 864 seconds by the end of the day.

Scenario	Change in Period (T)	Clock Status	Result
Length (L) Increases	T Increases	Runs Slow	Loses Time
Length (L) Decreases	T Decreases	Runs Fast	Gains Time
Gravity (g) Decreases	T Increases	Runs Slow	Loses Time

Sources: Science, Class VIII (NCERT), Keeping Time with the Skies, p.178; Science, Class VII (NCERT), Measurement of Time and Motion, p.108; Science, Class VII (NCERT), Measurement of Time and Motion, p.110

8. Solving the Original PYQ (exam-level)

Now that you have mastered the Simple Pendulum formula and the Calculus of Small Changes, you can see how these building blocks converge in this classic UPSC problem. The core concept is the proportionality where the time period (T) is directly proportional to the square root of the length (L). As you learned in the General Science Handbook, for small percentage increases, the change in the result is approximately the power multiplied by the change in the variable. Since the power of L is 1/2 (due to the square root), a 2% increase in length translates directly to a 1% increase in the time period.

To solve this, reason through the mechanics of timekeeping: if the time period increases by 1%, every "tick" of the clock takes 1% longer than it should. This means that over the course of a full day, the clock will lag behind by exactly 1% of the total seconds available. Since a standard day contains 86,400 seconds (24 hours × 60 minutes × 60 seconds), we simply calculate 1% of 86,400. This gives us 864 s, which confirms that (D) is the only logically sound answer. This step-by-step connection between proportionality and total duration is a hallmark of UPSC physics problems.

Be careful not to fall for the common traps reflected in the other options. Option (C) 1728 s is a 2% change, which occurs if you forget to apply the square root rule and assume a linear relationship between length and time. Option (B) 3456 s is a 4% change, often chosen by students who accidentally double the percentage instead of halving it. Finally, (A) 3600 s is a "distractor" number that looks familiar because it represents the seconds in a single hour, but it has no mathematical basis in this specific calculation. Success here depends on consistent application of the power rule for errors and approximations.