The simple harmonic motion of a particle is given by y = 3 sing co t + 4 cos co t. Which one of the following is the amplitude of such motion?

1. Periodic and Oscillatory Motion (basic)

In the world of physics, motion that repeats itself at regular intervals of time is called Periodic Motion. A familiar example of this is the rotation of the Earth on its axis or its revolution around the Sun Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267. These movements occur with such precision that we use them to define our days and years. However, when we dive deeper into waves and acoustics, we focus on a specific variety of periodic motion known as Oscillatory Motion.

Oscillatory Motion (or vibration) occurs when an object moves to and fro about a fixed, central point called the mean position. Think of a simple pendulum: it consists of a small metallic ball, called a bob, hanging from a rigid support Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109. When you pull the bob to one side and release it, it doesn't just fall; it swings back and forth across its starting point. This repetitive "back and forth" path is the defining characteristic of an oscillation.

To measure these movements, we look at the Time Period, which is the time taken to complete one full oscillation Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118. For a pendulum or a playground swing, this time period is remarkably consistent for a given length, regardless of how heavy the person on the swing is Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119. It is crucial to remember that while every oscillation is periodic (because it repeats), not every periodic motion is an oscillation. For example, the hands of a clock move periodically, but they don't swing "to and fro," so they are periodic but not oscillatory.

Feature	Periodic Motion	Oscillatory Motion
Movement	Repeats at regular intervals.	Repeats "to and fro" about a mean position.
Nature	Broad category.	A specific type of periodic motion.
Example	Orbit of planets, rotation of Earth.	Vibrating guitar string, simple pendulum.

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 VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.119; Physical Geography by PMF IAS, The Motions of The Earth and Their Effects, p.267

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

To understand the complex world of waves and acoustics, we must first master Simple Harmonic Motion (SHM). At its heart, SHM is a specific type of periodic motion—motion that repeats itself at regular intervals of time. Think of a simple pendulum: a metallic ball (the bob) suspended by a thread. When at rest, it sits at its mean position; when nudged, it oscillates back and forth Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.109. The time it takes to complete one full back-and-forth swing is known as the time period (T), which remains constant for a pendulum of a fixed length Science-Class VII . NCERT(Revised ed 2025), Measurement of Time and Motion, p.118.
What makes SHM 'simple' is that the restoring force (the force pulling the object back to the center) is directly proportional to the displacement (how far it has moved from the center). The maximum displacement from the mean position is called the amplitude (A). In mathematical terms, we represent this motion using sine or cosine functions because they naturally repeat their values. For example, a particle's position might be described as y = A sin(ωt), where ω (omega) represents the angular frequency. This motion can be visualized as a wave creating peaks (crests) and valleys (troughs) Fundamentals of Physical Geography, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20.
Often, a particle is subjected to two different harmonic influences simultaneously. When these motions occur along the same line or perpendicularly, they combine to form a resultant motion. If we combine a sine wave and a cosine wave of the same frequency (like y = A sin ωt + B cos ωt), they are effectively 90° out of phase. Because of this right-angle relationship in their 'phase,' we calculate the resultant amplitude (R) using the Pythagorean theorem: R = √(A² + B²). This is a fundamental principle in superposition, allowing us to simplify complex wave interactions into a single, measurable motion.

Term	Definition	Significance
Mean Position	The equilibrium point where the object rests.	The center point of oscillation.
Amplitude	Maximum displacement from the mean position.	Determines the 'intensity' or energy of the motion.
Time Period	Time taken for one complete oscillation.	Determines the frequency (f = 1/T).

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; Fundamentals of Physical Geography, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20

3. Wave Parameters: Amplitude, Phase, and Frequency (intermediate)

To understand waves, we must look beyond the surface and identify the specific parameters that define their behavior. At its core, a wave is a disturbance that carries energy. The most fundamental parameter is Amplitude. In physical terms, such as ocean waves, we define Wave Height as the vertical distance from the bottom of a trough to the top of a crest. The Amplitude is exactly one-half of this wave height FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109. Mathematically, amplitude represents the maximum displacement of a particle from its equilibrium (rest) position. When multiple waves or oscillations combine—such as a sine wave and a cosine wave of the same frequency—the resultant amplitude is not a simple sum. Because sine and cosine are 90 degrees (π/2) out of phase, we calculate the total amplitude (R) using the Pythagorean theorem: R = √(A² + B²), where A and B are the individual coefficients.

The temporal aspects of a wave are defined by Frequency and Period. Wave Period is the time interval required for two successive crests or troughs to pass a fixed point Physical Geography by PMF IAS, Tsunami, p.192. Conversely, Wave Frequency is the number of waves passing a point in one second. These two are inversely related (f = 1/T). In a Simple Harmonic Motion (SHM) equation like y = A sin(ωt), the term ω (omega) represents the angular frequency, which dictates how fast the wave oscillates through its cycle.

Finally, we must consider Phase. Phase describes the specific location of a point within a wave cycle at a specific time (t=0). If two waves reach their peaks at the same time, they are "in phase." If one is at a peak while the other is at a trough, they are 180° "out of phase." Understanding phase is critical because it determines how waves interfere with one another. A practical application of these parameters is seen in the Shoaling Effect: as a tsunami enters shallow water, its wavelength decreases but its Amplitude increases significantly due to the conservation of energy, often rising from a negligible height in the deep ocean to 20 or 30 meters at the coast Physical Geography by PMF IAS, Tsunami, p.193.

Parameter	Definition	UPSC Significance
Amplitude	Max displacement from equilibrium (1/2 wave height).	Determines the energy and intensity of the wave (e.g., Tsunami height).
Frequency	Cycles per second (Hertz).	Determines the nature of the wave (e.g., light color or sound pitch).
Phase	Position within the cycle at a specific time.	Explains interference patterns and combining multiple oscillations.

Sources: FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Movements of Ocean Water, p.109; Physical Geography by PMF IAS, Tsunami, p.192-193

4. Classification of Waves: Transverse and Longitudinal (basic)

To understand waves, we must look at how the particles of a medium move relative to the direction the energy is traveling. This gives us two primary categories: Transverse and Longitudinal waves. In a Transverse wave, the particles of the medium vibrate perpendicularly (at 90°) to the direction of wave propagation. Think of a rope tied to a wall; when you shake it up and down, the wave moves toward the wall, but the rope itself moves up and down. These waves create crests (peaks) and troughs (valleys) as they distort the medium. Common examples include light waves and S-waves (Secondary waves) during an earthquake Physical Geography by PMF IAS, Earths Interior, p.62.

In contrast, Longitudinal waves are those where the particles of the medium vibrate parallel to the direction of the wave's travel. Instead of crests and troughs, these waves move through a series of compressions (high-pressure zones) and rarefactions (low-pressure zones). Sound is the most famous example of a longitudinal mechanical wave Physical Geography by PMF IAS, Earths Magnetic Field, p.64. Similarly, in seismology, P-waves (Primary waves) are longitudinal; they push and pull the rock in the same direction the wave travels, making them significantly faster than transverse waves FUNDAMENTALS OF PHYSICAL GEOGRAPHY (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20.

Feature	Transverse Waves	Longitudinal Waves
Particle Motion	Perpendicular to wave direction	Parallel to wave direction
Formed By	Crests and Troughs	Compressions and Rarefactions
Examples	Light, S-waves, Water ripples	Sound, P-waves, Ultrasound

Sources: Physical Geography by PMF IAS, Earths Interior, p.62; Physical Geography by PMF IAS, Earths Magnetic Field (Geomagnetic Field), p.64; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20

5. Phenomenon of Interference and Superposition (intermediate)

In the study of physics, the Principle of Superposition is a fundamental rule that explains what happens when two or more waves overlap at the same point in space. Unlike solid objects that collide and bounce off one another, waves pass through each other, and their effects simply add up. At any given moment, the resultant displacement of the medium is the vector sum of the individual displacements caused by each wave. This principle applies to all types of waves, whether they are the longitudinal P-waves (primary waves) that compress and stretch the ground, or the transverse S-waves (secondary waves) that create crests and troughs Physical Geography by PMF IAS, Earths Interior, p.60-62.

Interference is the specific phenomenon resulting from this superposition when waves of the same frequency meet. When two waves arrive in step (in phase), they reinforce each other, leading to constructive interference. Conversely, if they arrive out of step, they can partially or fully cancel each other out, known as destructive interference. In a laboratory or mathematical setting, we often analyze this by combining two different simple harmonic motions (SHM). For instance, if a particle is influenced by a sine function and a cosine function of the same frequency, we are essentially looking at two oscillations that are exactly 90 degrees (π/2) out of phase.

To find the resultant amplitude of such a combined motion, we use a method called phasor addition. Since the sine and cosine components are perpendicular in their phase relationship, they can be treated like the legs of a right-angled triangle. Applying the Pythagorean theorem, the resultant amplitude (R) is calculated as R = √(A² + B²), where A and B are the coefficients (amplitudes) of the sine and cosine terms. This elegant mathematical bridge allows us to predict the intensity of sound, the brightness of light, or the magnitude of seismic vibrations when multiple wave sources interact FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20.

Sources: Physical Geography by PMF IAS, Earths Interior, p.60-62; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), The Origin and Evolution of the Earth, p.20

6. Seismic Waves and Sound Applications (exam-level)

To understand the Earth's interior and the physics of acoustics, we must look at seismic waves. These are mechanical waves of energy that travel through the Earth, triggered by sudden breaks in rock or explosions. They function much like high-energy sound waves, acting as a "CT scan" for our planet. We categorize these primarily into Body waves (which travel through the Earth's interior) and Surface waves (which move along the surface). Fundamentals of Physical Geography, The Origin and Evolution of the Earth, p.20.

Body waves are further divided into P-waves and S-waves, and their physical behavior tells us everything we know about what lies beneath our feet:

Feature	P-waves (Primary)	S-waves (Secondary)
Wave Type	Longitudinal (Compressional). Particles move parallel to the wave direction.	Transverse (Shear). Particles move perpendicular to the wave direction.
Medium	Travel through solids, liquids, and gases.	Travel only through solids.
Speed	Fastest (approx. 1.7 times faster than S-waves).	Slower; arrive after P-waves.
Sound Analogy	Analogous to sound waves in air.	Analogous to light waves or ripples in water.

The physics of these waves is dictated by the medium's properties. For instance, the velocity of seismic waves increases as the density of the material increases. Fundamentals of Physical Geography, The Origin and Evolution of the Earth, p.20. When these waves hit a boundary between different layers (like the crust and mantle), they undergo reflection (bouncing back) and refraction (bending). Physical Geography by PMF IAS, Earths Interior, p.61.

While P and S waves are vital for scientific mapping, Surface waves are the last to be recorded on a seismograph but are by far the most destructive. They cause the actual displacement of the ground that we feel during an earthquake. Physical Geography by PMF IAS, Earths Interior, p.62.

Sources: Fundamentals of Physical Geography, The Origin and Evolution of the Earth, p.20; Physical Geography by PMF IAS, Earths Interior, p.60; Physical Geography by PMF IAS, Earths Interior, p.61; Physical Geography by PMF IAS, Earths Interior, p.62

7. Resultant Amplitude of Combined Oscillations (exam-level)

Welcome to this crucial stage of our journey! To master wave motion, we must understand how individual oscillations combine to create a resultant motion. When a particle is subjected to two or more simple harmonic motions (SHM) simultaneously, we apply the Principle of Superposition. If these oscillations have the same frequency, the particle continues to move in SHM, but its amplitude—which is defined as one-half of the total wave height Physical Geography by PMF IAS, Tsunami, p.192—is determined by the relationship between the individual components.

Consider a motion expressed as y = A sin ωt + B cos ωt. At first glance, this looks like two different waves, but mathematically, it represents a single harmonic oscillation. Because the sine and cosine functions are exactly 90° (π/2 radians) out of phase with each other, we can treat their coefficients, A and B, as perpendicular components of a vector. This is often called the Phasor Method. Just as you would find the magnitude of a resultant vector using the Pythagorean theorem, the resultant amplitude (R) of this combined motion is calculated as the square root of the sum of the squares of the individual amplitudes.

The formula is straightforward: R = √(A² + B²). For example, if a motion is defined by the equation y = 3 sin ωt + 4 cos ωt, we identify A as 3 and B as 4. Plugging these into our formula gives R = √(3² + 4²) = √(9 + 16) = √25 = 5 units. This resultant amplitude represents the maximum displacement of the particle from its mean position during its combined vibration.

Sources: Physical Geography by PMF IAS, Tsunami, p.192

8. Solving the Original PYQ (exam-level)

Now that you have mastered the superposition principle and the behavior of phasors, this question serves as a direct application of those building blocks. The equation y = 3 sin ωt + 4 cos ωt represents a linear combination of two simple harmonic motions. Since sine and cosine functions are out of phase by exactly 90 degrees (π/2), they can be treated as orthogonal vectors. In your recent lessons, you learned that when two waves of the same frequency are combined, the resulting motion is also simple harmonic, and its magnitude is determined by the vector sum of their individual amplitudes.

To arrive at the correct answer, you should visualize this as a right-angled triangle where the coefficients 3 and 4 are the two legs. Applying the resultant amplitude formula, R = √(A² + B²), we substitute the given values: √(3² + 4²) = √(9 + 16) = √25. This calculation yields an amplitude of (B) 5. This process mirrors the way we calculate the hypotenuse in geometry, translating physical oscillations into a manageable mathematical form as described in NCERT Physics Class XI - Oscillations.

UPSC often includes "distractor" options to test the depth of your conceptual clarity. Option (C) 7 is a linear addition trap for students who forget the phase difference and simply add 3 and 4. Option (A) 1 is a subtraction trap, assuming the waves might cancel each other out, while (D) 12 is a multiplication trap. By recognizing that sine and cosine are perpendicular phasors, you avoid these arithmetic errors and correctly identify the geometric relationship required to solve the problem.