It is impossible for two oscillators, each excuting simple harmonic motion, to remain in phase with each other if they have different

1. Understanding Periodic and Oscillatory Motion (basic)

Welcome to your first step in mastering mechanics! To understand how things move, we first distinguish between motion that happens once and motion that repeats. Periodic motion is any motion that repeats itself at regular, fixed intervals of time. A classic example is the Earth revolving around the Sun every 365 days or a train moving at a constant speed covering equal distances in equal time intervals, which we call uniform linear motion Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.117.

Oscillatory motion is a specific type of periodic motion. It is characterized by a "to-and-fro" or back-and-forth movement about a central point, known as the mean position. Think of a simple pendulum: a small metallic ball (the bob) hanging from a thread. When at rest, it sits at its mean position. When you pull it to one side and release it, it oscillates between two extreme points Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.109. While all oscillatory motions are periodic (because they repeat their path in fixed times), not all periodic motions are oscillatory. For instance, the Earth orbits the Sun periodically, but it doesn't move "back and forth" over the same path.

Feature	Periodic Motion	Oscillatory Motion
Definition	Repeats at regular intervals.	Back-and-forth movement about a mean position.
Path	Can be circular, linear, or complex.	Must follow the same path to-and-fro.
Example	Hands of a clock.	A swinging cradle or a pendulum.

The most vital measurement in this context is the Time Period (T). This is the time taken by the object to complete one full oscillation (moving from the mean position to one extreme, to the other extreme, and back to the mean). For a pendulum of a specific length, this time period remains constant Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.118. If two oscillators have the same time period, they are said to be "in step" or in phase. If their time periods differ, they will eventually drift apart in their timing, even if they started at the same moment.

Sources: Science-Class VII . NCERT(Revised ed 2025), 8: Measurement of Time and Motion, p.109, 117-118

2. Simple Harmonic Motion (SHM) Fundamentals (basic)

To understand Simple Harmonic Motion (SHM), we must first look at the simplest example: the **pendulum**. A simple pendulum consists of a small metallic bob suspended by a thread. When it is at rest, it sits at its **mean position**. Once displaced and released, it begins an oscillatory motion that is **periodic**—meaning it repeats its path after a fixed interval of time Science-Class VII, Chapter 8, p.109. This fixed interval is known as the Time Period (T), and for a pendulum of a specific length, this period remains constant Science-Class VII, Chapter 8, p.118.

The 'status' of this vibration at any given moment is called its Phase. Mathematically, we describe the phase using the expression (ωt + φ). Here, ω (angular frequency) represents how fast the oscillation occurs (calculated as ω = 2π/T), and φ (phase constant) tells us the starting position of the oscillator at time t=0. If you have two different oscillators, they are said to be 'in phase' if they reach their peak heights and pass through their mean positions at the exact same time.

For two oscillators to maintain a constant phase relationship (staying 'in sync' over time), they must have identical Time Periods. If their time periods differ, their angular frequencies (ω) will also differ. This causes the term 'ωt' to grow at different rates for each oscillator, leading to a phase difference that continuously increases or decreases. While factors like amplitude (how far it swings) or energy affect the intensity of the motion, they do not dictate whether two systems stay in step—that is the job of the frequency.

Term	Definition	Role in SHM
Time Period (T)	Time taken for one full oscillation.	Determines the 'tempo' of the motion.
Phase (ωt + φ)	The state of motion at time 't'.	Describes the position and direction of the bob.
Angular Frequency (ω)	Rate of change of phase.	Connects time to the physical cycle (2π/T).

Sources: Science-Class VII, Measurement of Time and Motion, p.109; Science-Class VII, Measurement of Time and Motion, p.118; Fundamentals of Physical Geography, Geography Class XI, The Origin and Evolution of the Earth, p.20

3. Key Variables: Amplitude, Time Period, and Frequency (basic)

To understand the mechanics of any repeating motion—whether it is a pendulum swinging in a clock or a giant tsunami wave traveling across the ocean—we must master three fundamental variables: Amplitude, Time Period, and Frequency. These terms describe the 'size,' the 'timing,' and the 'pace' of the motion. Let us start with the Time Period (T). In a simple pendulum, one full oscillation occurs when the bob moves from its center (mean) position to one extreme, then to the other extreme, and finally returns to the center. The time taken for this single complete cycle is the time period Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.109. A key characteristic of a stable pendulum is that its time period remains remarkably consistent across successive swings, provided the length of the string doesn't change Science-Class VII . NCERT(Revised ed 2025), Chapter 8, p.110.

While the time period tells us how long one cycle takes, Frequency (f) tells us how many cycles happen in a unit of time (usually one second). They are two sides of the same coin: if a wave takes a long time to complete one cycle (high time period), fewer waves will pass by in a second (low frequency). Mathematically, they are inversely related: f = 1/T. In the context of oceanography, wave frequency is defined as the number of wave crests passing a fixed point in one second Fundamentals of Physical Geography, Geography Class XI (NCERT 2025 ed.), Chapter 13, p.109. Similarly, the term Amplitude measures the 'intensity' or 'displacement' of the motion. For a wave, the amplitude is exactly one-half of the total vertical distance between the highest point (crest) and the lowest point (trough) Physical Geography by PMF IAS, Tsunami, p.192.

Variable	Core Definition	Simple Analogy
Amplitude	Maximum displacement from the equilibrium (half the wave height).	How high the swing goes.
Time Period	Time taken to complete one full cycle/oscillation.	How long one full back-and-forth swing takes.
Frequency	Number of cycles completed in one second.	How fast the rhythm of the swings is.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.109-110; Fundamentals of Physical Geography, Geography Class XI (NCERT 2025 ed.), Chapter 13: Movements of Ocean Water, p.109; Physical Geography by PMF IAS, Tsunami, p.192

4. The Simple Pendulum and Laws of Oscillation (intermediate)

A simple pendulum is a quintessential model used in physics to understand periodic motion. It consists of a small, heavy mass called a bob, suspended by a light, inextensible string from a rigid support. When the bob is displaced slightly and released, it begins a back-and-forth movement known as oscillatory motion. This motion is considered a type of Simple Harmonic Motion (SHM) when the displacement is small.

To master this concept, we must first define what constitutes "one oscillation." An oscillation is completed when the bob moves from its mean position (center) to one extreme position, then to the other extreme, and finally returns to the center. Alternatively, it is the travel from one extreme to the other and back again. The time taken to complete this full cycle is known as the time period (T), and its SI unit is the second (s) Science-Class VII . NCERT (Revised ed 2025), Chapter 8, p. 109, 118.

The behavior of a pendulum is governed by specific laws that are often counter-intuitive. Many students assume that a heavier bob will swing faster, but this is a misconception. Let's look at the factors that actually influence the time period:

• Length of the String: The time period is directly proportional to the square root of the length (L). If you increase the length, the pendulum swings slower.
• Mass of the Bob: Surprisingly, the time period does not depend on the mass of the bob. A lead ball and a wooden ball on strings of equal length will have identical time periods Science-Class VII . NCERT (Revised ed 2025), Chapter 8, p. 110.
• Acceleration due to Gravity: The time period depends on the local gravity (g). This is why a pendulum clock that is accurate at sea level might lose time on a high mountain where gravity is slightly weaker.

When we move into more intermediate mechanics, we talk about the phase of the oscillation. The phase represents the state of the pendulum's motion at any given time t, expressed mathematically through the angular frequency (ω). For two pendulums to stay "in sync" (maintain a constant phase relationship), they must have identical angular frequencies. Since ω = 2π/T, this means they must have the same time period. If their time periods differ even slightly, their phases will drift apart over time, and they will eventually swing in opposite directions.

Sources: Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.109; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.110; Science-Class VII . NCERT(Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.118

5. Resonance and Forced Oscillations (intermediate)

Every physical system, from a simple pendulum to a massive bridge, has a natural frequency—the specific rate at which it vibrates when disturbed and left to itself. When we apply an external, periodic force to such a system, we create forced oscillations. In this state, the system is 'forced' to vibrate at the frequency of the external driver rather than its own natural rhythm. For example, in the ionosphere, high-frequency radio waves act as an external force that hits free electrons, causing them to vibrate and re-radiate energy Physical Geography by PMF IAS, Earths Atmosphere, p.279.

Resonance is a special and powerful case of forced oscillation. It occurs when the frequency of the external periodic force exactly matches the natural frequency of the system. When this 'sync' happens, the system absorbs energy with maximum efficiency, leading to a dramatic increase in the amplitude of the oscillation. This principle is why a singer can shatter a glass by hitting a specific note or why musical instruments use ductile metal wires tuned to specific frequencies to produce rich, loud sounds Science-Class VII . NCERT(Revised ed 2025), The World of Metals and Non-metals, p.44.

Type of Oscillation	Driving Force	Frequency of Motion
Free Oscillation	None (Initial disturbance only)	Natural frequency of the system.
Forced Oscillation	External periodic force	Frequency of the external force.
Resonance	External force matches natural frequency	Natural frequency (resulting in max amplitude).

In the real world, resonance can be both a tool and a hazard. Engineers must ensure that the natural frequency of a building does not match the frequency of typical earthquake waves, as resonance could cause the structure to shake violently and collapse Physical Geography by PMF IAS, Earthquakes, p.182. Similarly, large bodies of water can experience seiches—a pendulum-like rocking—when external forces like wind or tides excite the lake's resonant frequency, causing the water level to rise and fall at regular intervals Environment and Ecology, Majid Hussain, Natural Hazards and Disaster Management, p.58.

Sources: Physical Geography by PMF IAS, Earths Atmosphere, p.279; Science-Class VII . NCERT(Revised ed 2025), The World of Metals and Non-metals, p.44; Physical Geography by PMF IAS, Earthquakes, p.182; Environment and Ecology, Majid Hussain, Natural Hazards and Disaster Management, p.58

6. The Concept of Phase and Phase Difference (exam-level)

In the study of periodic motion, such as a swinging pendulum or a vibrating string, the term phase describes the specific state of an oscillator at a given moment. Think of it as a status report that tells us two things: the object's position and its direction of motion. While we often talk about a pendulum's motion in terms of time, as seen in Science-Class VII, NCERT (Revised ed 2025), Chapter 8: Measurement of Time and Motion, p. 110, the mathematical description of this state is expressed as an angle, usually in the form (ωt + φ). Here, ω represents the angular frequency, t is time, and φ is the phase constant (the starting position at t=0).

When comparing two different oscillators, we look at their phase difference. If two pendulums reach their highest point at the exact same moment, they are said to be "in phase." If one is at its peak while the other is at its lowest point, they are 180 degrees (or π radians) "out of phase." Interestingly, the term "phase" is also used in physical geography to describe changes in the state of matter, where a substance transitions (e.g., from liquid to gas) without a change in temperature, as discussed in Physical Geography by PMF IAS, Vertical Distribution of Temperature, p. 295. However, in mechanics, we are specifically looking at the timing of cycles.

A crucial concept for your exam is understanding what allows two oscillators to maintain a constant phase relationship over time. For the phase difference to remain the same, both oscillators must complete their cycles at the same rate. This means they must have identical time periods (T) and, consequently, identical angular frequencies (ω = 2π/T). If one pendulum swings slightly faster than the other, their phase difference will keep increasing or decreasing, causing them to drift "out of sync" regardless of whether they started together or have the same amplitude.

Sources: Science-Class VII, NCERT (Revised ed 2025), Chapter 8: Measurement of Time and Motion, p.110; Physical Geography by PMF IAS, Vertical Distribution of Temperature, p.295

7. Angular Frequency and Phase Evolution (exam-level)

In the study of oscillatory motion, the phase of an object represents its specific position and direction of motion at any given moment. Mathematically, for a system undergoing Simple Harmonic Motion (SHM), the phase at time 't' is expressed as (ωt + φ). Here, ω (omega) is the angular frequency, and φ (phi) is the phase constant or initial phase. The angular frequency is fundamentally linked to the time period (T) by the relationship ω = 2π/T. This value tells us how rapidly the phase of the system evolves as time passes.

For two oscillators to stay in phase—meaning they maintain a constant relative position in their cycles—their phase difference must remain constant over time. If we have two pendulums with different time periods, they will naturally have different angular frequencies (ω₁ and ω₂). Consequently, the term 'ωt' will grow at different rates for each pendulum. Even if they start exactly together, the one with the higher frequency will "gain" phase faster, leading to a continuously increasing phase difference. This is why having identical time periods is the essential condition for maintaining a constant phase relationship; factors like amplitude or the specific energy of the system do not dictate whether the oscillators stay in step.

We see these principles of periods and phases reflected in celestial mechanics as well. For instance, the synodic month (29.53 days) is defined as the time between successive recurrences of the same phase of the Moon, such as from one full moon to the next Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.261. Similarly, the Earth possesses a constant angular velocity (ω) as it rotates, a factor that is critical in calculating the Coriolis force (2νω sin ϕ) experienced by moving objects across different latitudes Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309.

Sources: Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.261; Physical Geography by PMF IAS, Pressure Systems and Wind System, p.309

8. Solving the Original PYQ (exam-level)

Now that you have mastered the fundamental mechanics of a simple pendulum and the governing equations of Simple Harmonic Motion (SHM), this question tests your ability to connect the "timing" of an oscillator to its mathematical phase. In your recent lessons, you learned that phase represents the position and direction of an oscillator at any given moment. For two oscillators to remain in phase, they must stay perfectly synchronized over time, meaning they must complete their cycles at the exact same rate. This rate is determined by the angular frequency (ω), which is inversely proportional to the time period (T) through the relationship ω = 2π/T.

To arrive at the correct answer, think of phase as a clock. If two clocks run at different speeds, they will eventually show different times, regardless of where they started. Mathematically, the phase is expressed as (ωt + φ); if the time periods differ, the ω term differs, causing the phase difference between the two oscillators to increase or decrease continuously as time (t) progresses. Therefore, it is impossible for them to remain synchronized if they have different (A) time periods. As discussed in Science-Class VII . NCERT(Revised ed 2025), the constancy of a pendulum's period is what allows for precise timekeeping.

UPSC often uses amplitudes and kinetic energy as distractors to see if you confuse the intensity of motion with its timing. Two pendulums can swing at different widths (amplitudes) or carry different levels of energy, yet still cross the equilibrium point at the same time if their periods are identical. Similarly, while spring constants influence the period, they are a physical property of the hardware; two different springs can still be in phase if their masses are adjusted to result in the same period. The trap is thinking physical differences always disrupt phase, but only a difference in the time period makes phase synchronization fundamentally impossible.