Question

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

Hint:

Whenever you see a combination of sine and cosine functions with the same frequency, like \(a \sin(\omega t) + b \cos(\omega t)\), immediately think of combining them into a single sine (or cosine) function. The new amplitude will always be \( \sqrt{a^2 + b^2} \).

Answer 1

Correct Answer: 1

Step 1: Understanding the Question:
We are given two equations of SHM. We need to find the amplitude of each motion and then find the ratio of their amplitudes to determine the value of x.
Step 2: Key Formula or Approach:
1. The standard form of an SHM equation is \(y = A \sin(\omega t + \phi)\), where A is the amplitude.
2. An expression of the form \(a \sin(\theta) + b \cos(\theta)\) can be converted to \(R \sin(\theta + \alpha)\), where the amplitude \(R = \sqrt{a^2 + b^2}\).
Step 3: Detailed Explanation:
Amplitude of \(y_1\):
The first equation is already in the standard form:
\(y_1 = 10 \sin(3\pi t + \pi/3)\)
By comparing with \(y = A \sin(\omega t + \phi)\), we can see that the amplitude of \(y_1\) is \(A_1 = 10\).
Amplitude of \(y_2\):
The second equation is:
\(y_2 = 5 (\sin(3\pi t) + \sqrt{3} \cos(3\pi t))\)
Let's first simplify the expression in the parenthesis: \( \sin(3\pi t) + \sqrt{3} \cos(3\pi t) \).
This is in the form \(a \sin\theta + b \cos\theta\) with \(a=1\), \(b=\sqrt{3}\), and \(\theta = 3\pi t\).
The amplitude of this part is \( R = \sqrt{a^2 + b^2} = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1+3} = \sqrt{4} = 2 \).
So, \( \sin(3\pi t) + \sqrt{3} \cos(3\pi t) = 2 \sin(3\pi t + \alpha) \) for some phase \(\alpha\).
Now, substitute this back into the equation for \(y_2\):
\( y_2 = 5 \times [2 \sin(3\pi t + \alpha)] = 10 \sin(3\pi t + \alpha) \).
The amplitude of \(y_2\) is \(A_2 = 10\).
Ratio of Amplitudes:
The ratio of the amplitude of \(y_1\) to \(y_2\) is:
\[ \frac{A_1}{A_2} = \frac{10}{10} = 1 \] We are given that this ratio is \(x:1\), which means \( \frac{A_1}{A_2} = \frac{x}{1} \).
Comparing the two, we get \( x = 1 \).
Step 4: Final Answer:
The value of x is 1.