The displacement y of a particle executing periodic motion is given by y = 4 cos^2(t/2) sin(1000t).

Q: Why can't we just say there are two harmonic motions because of the terms $\cos^2(t/2)$ and $\sin(1000t)$?

A simple harmonic motion must be represented by a single sinusoidal term of the form $A \sin(\omega t + \phi)$. A product of two trigonometric functions represents a modulated wave, not a single SHM. Decomposing the product reveals the actual individual frequencies present in the superposition.

Q: What is the physical meaning of this decomposition?

This is the principle of Fourier analysis and Amplitude Modulation (AM). It shows that a modulated wave (where amplitude varies over time) is physically equivalent to the superposition of three constant-amplitude waves of slightly different frequencies (the carrier and two sidebands).

Q: Does the amplitude of the individual waves affect the number of independent motions?

No, the number of independent motions is determined solely by the number of distinct frequency components (terms with different $\omega$) in the fully expanded linear sum.

Q: What if the term was $\cos^3(t)$ instead of $\cos^2(t/2)$?

You would use the triple-angle identity $\cos(3t) = 4\cos^3(t) - 3\cos(t)$ to linearize it. The process remains the same: linearize all powers of trigonometric functions and then convert products to sums.

Q: Are the three resulting waves in phase with each other?

Yes, in this specific case, all three decomposed terms are sine functions with zero phase constant, meaning they all pass through zero at $t=0$.

Question

The displacement $y$ of a particle executing periodic motion is given by $y = 4 \cos^2\left(\frac{1}{2}t\right) \sin(1000t)$. This expression may be considered to be a result of the superposition of ........................ independent harmonic motions.

Options:

(A) two

(B) three

(C) four

(D) five

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are looking at a complex wave pattern on an oscilloscope.

At first glance, the equation representing this wave looks intimidating:

$$y = 4 \cos^2\left(\frac{1}{2}tight) \sin(1000t)$$

It is a product of a squared trigonometric function and a high-frequency sinusoidal function.

In physics, when waves superpose, they add up linearly.

Therefore, to find out how many **independent simple harmonic motions (SHMs)** make up this complex wave, we must decompose this product into a sum of individual sine or cosine terms.

Each term in the final sum of the form $A \sin(\omega t + \phi)$ will represent one independent harmonic motion.

### The Master Equation & Linearization

Our first obstacle is the squared term, $\cos^2\left(\frac{1}{2}tight)$.

To linearize this, we recall the standard double-angle identity for cosine:

$$\cos(2	heta) = 2 \cos^2(	heta) - 1$$

By rearranging this identity, we can express the squared term in a linear form:

$$2 \cos^2(	heta) = 1 + \cos(2	heta)$$

Let's apply this to our equation by setting $	heta = \frac{t}{2}$. This gives:

$$2 \cos^2\left(\frac{t}{2}ight) = 1 + \cos(t)$$

Now, we rewrite our original displacement equation to isolate this factor of $2$:

$$y = 2 \left[ 2 \cos^2\left(\frac{t}{2}ight) ight] \sin(1000t)$$

Substituting our linearized identity into the equation, we get:

$$y = 2 \left( 1 + \cos(t) ight) \sin(1000t)$$

Let's expand this expression by distributing the terms:

$$y = 2 \sin(1000t) + 2 \sin(1000t) \cos(t)$$

We have successfully broken the equation into two parts!

The first term, $2 \sin(1000t)$, is already a perfect simple harmonic wave with an angular frequency of $\omega = 1000	ext{ rad/s}$.

Now, we must tackle the second term, which is still a product: $2 \sin(1000t) \cos(t)$.

### Decomposing the Product Term

To split the product of a sine and a cosine into a sum, we use the **product-to-sum formula**:

$$2 \sin(A) \cos(B) = \sin(A + B) + \sin(A - B)$$

In our case, we identify $A = 1000t$ and $B = t$.

Substituting these values into the formula:

$$2 \sin(1000t) \cos(t) = \sin(1000t + t) + \sin(1000t - t)$$

Simplifying the angles inside the sine functions:

$$2 \sin(1000t) \cos(t) = \sin(1010t) + \sin(999t)$$

This is incredibly elegant! The single product term has split into two distinct, independent harmonic waves.

### Final Calculation & Physical Significance

Now, let's substitute this back into our expanded displacement equation:

$$y = 2 \sin(1000t) + \sin(1001t) + \sin(999t)$$

Look at this final expression! It is the sum of exactly **three independent simple harmonic motions**:

1. A wave of amplitude $2$ and angular frequency $\omega_2 = 1000	ext{ rad/s}$.
2. A wave of amplitude $1$ and angular frequency $\omega_3 = 1001	ext{ rad/s}$.
3. A wave of amplitude $1$ and angular frequency $\omega_1 = 999	ext{ rad/s}$.

This mathematical decomposition is not just a theoretical exercise; it is the exact physics behind **Amplitude Modulation (AM)** used in radio communication.

The high-frequency wave $\sin(1000t)$ is the **carrier wave**, and the slow-varying term $(1 + \cos(t))$ is the **modulating signal**.

The resulting frequencies ($999	ext{ rad/s}$, $1000	ext{ rad/s}$, and $1001	ext{ rad/s}$) are known as the **carrier** and the **sidebands**.

Thus, the correct option is **(b) three**.

Animated Solution for Physics - Waves: The displacement $y$ of a particle executing periodic motion is given by $y = 4 cos^{2} (\frac{1}{2} t) sin (1000 t)$ . This expression may be considered to be a result of the superposition of ........................ independent harmonic motions.

Select Answer:

Visualized Solution

Analyzing the Wave Equation

The Cosine Squared Identity

Substituting the Identity

Expanding the Expression

The Product-to-Sum Tool

Applying the Product-to-Sum Formula

The Final Decomposed Wave

Physical Significance: Amplitude Modulation

The Sigma Insight: Interference of Waves

Analyzing the Setup

The Master Equation & Linearization

Decomposing the Product Term

Final Calculation & Physical Significance

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