91Ó°ÊÓ

In Simple Harmonic Motion (SHM), the amplitude is the maximum extent of oscillation or the peak value attained by the wave. It is a crucial parameter as it indicates how much energy the wave potentially carries. The amplitude is depicted as a constant multiplier in the wave equation, which can be easily identified in the mathematical expression of SHM.

For example, in the equation of motion given by SHM, like for the wave \( Y_1 = 5 \sin(\omega t) \), the amplitude here is **5**. This means that the maximum value that this wave achieves above or below the equilibrium position is 5.

Similarly, for the wave \( Y_2 = 5[\sqrt{3} \sin(\omega t) + \cos(\omega t)] \), the amplitude isn’t as straightforward. We analyze and simplify this expression using trigonometric identities to ascertain that the amplitude is actually **15** after applying relevant formulas.

Trigonometric identities are fundamental tools in mathematics that help simplify expressions that involve angles and trigonometric functions like \( \sin \) and \( \cos \). In the context of our SHM problem, they are particularly useful for determining components that may not be immediately apparent.

One key identity that we use is \( \sin^2(\omega t) + \cos^2(\omega t) = 1 \). This identity helps us in transforming and simplifying expressions, especially when dealing with sum or difference inside a trigonometric function. For example, when given a complex SHM equation, breaking it down using this identity allows for precise calculation of terms involved.

In our problem with \( Y_2 = 5[\sqrt{3} \sin(\omega t) + \cos(\omega t)] \), the identity is crucial for determining the resultant amplitude, where it's used to simplify \( \sqrt{(\sqrt{3} \sin(\omega t))^2 + (\cos(\omega t))^2} = 3 \). Thus, trigonometric identities play an integral role in finding hidden amplitudes and simplifying wave expressions.

The ratio of amplitudes in SHM shows a comparative measure of maximum displacements for two different waves. This ratio doesn’t just provide a numerical relationship between two waveforms, but also insights into the energy and intensity they could potentially impart.

In our solved problem, we have two waves: \( Y_1 = 5\sin(\omega t) \) and \( Y_2 = 5[\sqrt{3}\sin(\omega t) + \cos(\omega t)] \). First, we identify their amplitudes: \( A_1 = 5 \) for \( Y_1 \), and after simplifying \( Y_2 \) using trigonometric identities, \( A_2 = 15 \).

We then compute the ratio of these amplitudes: \( \frac{A_1}{A_2} = \frac{5}{15} = \frac{1}{3} \). Therefore, the proportional relationship between these two harmonic motions is given by the ratio \( 1:3 \), indicating that the second wave has an amplitude three times larger than the first.