Two simple harmonic motion, are represented by the equations y1 = 10 sin(3πt + π/3) and y2 = 5(sin 3

Q: Can we convert $y_2$ into a single sine function?

Yes! By multiplying and dividing by the resultant amplitude (10), we get $y_2 = 10(\frac{1}{2} \sin 3\pi t + \frac{\sqrt{3}}{2} \cos 3\pi t) = 10 \sin(3\pi t + \pi/3)$. Notice it's exactly the same as $y_1$!

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

Then the resulting motion would not be a simple harmonic motion, and we couldn't find a single constant amplitude using this simple vector method. It would be a complex periodic motion.

Q: Does the phase constant $\pi/3$ in $y_1$ affect its amplitude?

No, the phase constant only shifts the wave in time (left or right on a graph). It does not change the maximum displacement, which is the amplitude.

Q: What is the physical meaning of adding sine and cosine terms?

It represents a single SHM that starts at some intermediate position and velocity at $t=0$, rather than starting exactly at the mean position (pure sine) or extreme position (pure cosine).

Question

Two simple harmonic motion, are represented by the equations
$y_1 = 10 \sin \left( 3\pi t + \frac{\pi}{3} \right)$
and $y_2 = 5(\sin 3\pi t + \sqrt{3} \cos 3\pi t)$
Ratio of amplitude of $y_1$ to $y_2 = x : 1$. The value of $x$ is ............ .

Sigma Percentile · Accepted Answer

### The Magic of Superposition in SHM

Simple Harmonic Motion (SHM) is one of the most elegant concepts in physics. Often, we encounter motions that are not just a simple sine wave, but a combination of sine and cosine waves. This problem is a beautiful demonstration of how we can decode such combinations.

### Analyzing the First Equation

Let's start with the first equation given to us:

$$y_1 = 10 \sin \left( 3\pi t + \frac{\pi}{3} \right)$$

This equation is already in the standard form of an SHM, which is $y = A \sin(\omega t + \phi)$. By directly comparing the two, we can immediately see that the amplitude of the first motion is:

$$A_1 = 10$$

### Decoding the Second Equation

The second equation looks a bit more complex because it mixes sine and cosine terms:

$$y_2 = 5(\sin 3\pi t + \sqrt{3} \cos 3\pi t)$$

To make sense of this, let's distribute the $5$ inside the bracket:

$$y_2 = 5 \sin 3\pi t + 5\sqrt{3} \cos 3\pi t$$

How do we find the amplitude of a motion that is a sum of a sine and a cosine? The secret lies in **phasors**. Because sine and cosine functions have a phase difference of exactly $90^\circ$ (or $\pi/2$ radians), we can treat their coefficients as perpendicular vectors.

### The Phasor Approach

Imagine a coordinate system where the horizontal axis represents the sine component and the vertical axis represents the cosine component.

Our horizontal vector has a length of $a = 5$, and our vertical vector has a length of $b = 5\sqrt{3}$. The resultant amplitude $A_2$ is simply the magnitude of the vector sum of these two components. Using the Pythagorean theorem:

$$A_2 = \sqrt{a^2 + b^2}$$

Substituting our values:

$$A_2 = \sqrt{5^2 + (5\sqrt{3})^2}$$

$$A_2 = \sqrt{25 + 75}$$

$$A_2 = \sqrt{100} = 10$$

### The Final Conclusion

Fascinatingly, both motions have the exact same amplitude! The ratio of their amplitudes is:

$$\frac{A_1}{A_2} = \frac{10}{10} = 1$$

The problem states that this ratio is $x : 1$. Therefore, comparing the two, we find our final answer:

**$$x = 1$$**

This problem teaches us a valuable lesson: a combination of sine and cosine waves of the same frequency is just another simple harmonic motion, shifted in phase, with an amplitude that can be easily found using vector addition.

Animated Solution for Physics - Oscillations: Two simple harmonic motion, are represented by the equations $y_{1} = 10 sin (3 π t + \frac{π}{3})$ and $y_{2} = 5 (sin 3 π t + 3 cos 3 π t)$ Ratio of amplitude of $y_{1}$ to $y_{2} = x : 1$ . The value of $x$ is ............ .

Enter Numerical Value:

Visualized Solution

Amplitude of First SHM

Expanding the Second SHM

Phasor Components

Resultant Amplitude

Final Ratio

The Sigma Insight: Simple Harmonic Motion (SHM)

The Magic of Superposition in SHM

Analyzing the First Equation

Decoding the Second Equation

The Phasor Approach

The Final Conclusion

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