The displacement of a particle varies according to the relation x = 4(cos(pi t) + sin(pi t)). The am

Q: Why can we treat sine and cosine as perpendicular vectors?

Because $\cos(\omega t) = \sin(\omega t + \pi/2)$. The phase difference of $\pi/2$ radians corresponds to a $90^\circ$ angle in the phasor diagram, making them perpendicular.

Q: Can I just add the amplitudes directly to get 8?

No! Amplitudes of SHMs must be added as vectors, taking their phase difference into account. Direct addition is only valid if they are perfectly in phase (phase difference of zero).

Q: What if the frequencies of the sine and cosine terms were different?

If the frequencies were different, the resulting motion would not be a simple harmonic motion. It would be a complex periodic motion, and you couldn't define a single constant amplitude in the same way.

Q: How do I find the initial phase angle of the resultant motion?

The phase angle $\phi$ can be found using $\tan\phi = \frac{A_2}{A_1}$, where $A_2$ is the amplitude of the cosine term and $A_1$ is the amplitude of the sine term. Here, $\tan\phi = 4/4 = 1$, so $\phi = \pi/4$.

Q: Is the phasor method always faster than the trigonometric method?

Usually, yes! For finding just the resultant amplitude or phase of superposed SHMs of the same frequency, drawing a quick right-angled triangle is much faster and less prone to algebraic errors than expanding trigonometric identities.

Question

The displacement of a particle varies according to the relation $x = 4(\cos\pi t + \sin\pi t)$. The amplitude of the particle is

Options:

(A) $-4$

(B) $4$

(C) $4\sqrt{2}$

(D) $8$

Sigma Percentile · Accepted Answer

### The Superposition of Simple Harmonic Motions

Imagine a particle being pulled by two different invisible springs at the same time. One spring pulls it according to a sine wave, and the other pulls it according to a cosine wave. This is exactly what the equation $x = 4(\cos\pi t + \sin\pi t)$ represents!

When a particle is subjected to two or more simple harmonic motions simultaneously, its resultant motion is the vector sum of the individual motions. This principle is known as the **superposition of SHMs**.

### The Phasor Method: A Visual Shortcut

While we could use trigonometric identities to solve this, there is a much more elegant and visual way: the **Phasor Method**.

In the world of phasors, we treat oscillating quantities as rotating vectors. The length of the vector represents the amplitude, and the angle represents the phase.

Because a cosine function is simply a sine function shifted by exactly $90^\circ$ (or $\frac{\pi}{2}$ radians), we can represent the sine term and the cosine term as two vectors that are perfectly perpendicular to each other.

### Calculating the Resultant Amplitude

Let's break down our equation:
$$x = 4\sin\pi t + 4\cos\pi t$$

We have two perpendicular phasors:
1. A vector of length $4$ along the x-axis (representing the sine term).
2. A vector of length $4$ along the y-axis (representing the cosine term).

To find the resultant amplitude, we simply need to find the vector sum of these two phasors. Since they form a right-angled triangle, we can call upon our old friend, the **Pythagoras theorem**.

$$A = \sqrt{A_1^2 + A_2^2}$$

Substituting our values:
$$A = \sqrt{4^2 + 4^2}$$
$$A = \sqrt{16 + 16}$$
$$A = \sqrt{32}$$

Simplifying the square root, we get our final answer:
$$A = 4\sqrt{2}$$

### The Trigonometric Alternative

For those who love pure algebra, we can also solve this by manipulating the trigonometric expression.

We multiply and divide the entire equation by $\sqrt{4^2 + 4^2} = 4\sqrt{2}$. Actually, it's easier to just factor out the $4$ and then multiply and divide by $\sqrt{1^2 + 1^2} = \sqrt{2}$.

$$x = 4\sqrt{2} \left( \frac{1}{\sqrt{2}}\sin\pi t + \frac{1}{\sqrt{2}}\cos\pi t \right)$$

Recognizing that $\cos\frac{\pi}{4} = \frac{1}{\sqrt{2}}$ and $\sin\frac{\pi}{4} = \frac{1}{\sqrt{2}}$, we can rewrite this as:
$$x = 4\sqrt{2} \left( \sin\pi t \cos\frac{\pi}{4} + \cos\pi t \sin\frac{\pi}{4} \right)$$

Using the compound angle formula $\sin(A + B) = \sin A \cos B + \cos A \sin B$, we get:
$$x = 4\sqrt{2} \sin\left(\pi t + \frac{\pi}{4}\right)$$

The coefficient in front of the sine function is our resultant amplitude, which confirms our previous result: **$4\sqrt{2}$**. Both methods are incredibly powerful, and mastering them will give you a significant edge in JEE Physics!

Animated Solution for Physics - Oscillations: The displacement of a particle varies according to the relation $x = 4 (cos π t + sin π t)$ . The amplitude of the particle is

Select Answer:

Visualized Solution

$x = 4 cos π t + 4 sin π t$

$Phasor Method$

$A_{1} = 4, A_{2} = 4$

$A = A_{1}^{2} + A_{2}^{2}$

$A = 42$

$x = 42 sin (π t + \frac{π}{4})$

The Sigma Insight: Simple Harmonic Motion (SHM)

The Superposition of Simple Harmonic Motions

The Phasor Method

Calculating the Resultant Amplitude

The Trigonometric Alternative

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Animated Solution for Physics - Oscillations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The displacement of a particle varies according to the relation x=4(cosπt+sinπt). The amplitude of the particle is

Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=4cosπt+4sinπt

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A1​=4,A2​=4

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A=A12​+A22​​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A=42​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x=42​sin(πt+4π​)