Question

Two simple harmonic motions are represented by \(y_1 = 5 \left[ \sin 2\pi t + \sqrt{3} \cos 2\pi t \right] \) and \(y_2 = 5 \sin \left[ 2\pi t + \frac{\pi}{4} \right] \). The ratio of their amplitudes is:

• A. 1:1
• B. 2:1
• C. 1:3
• D. \(\sqrt{3}:1\)

Hint:

When finding amplitudes of combined harmonic motions, use the identity \( A = \sqrt{a^2 + b^2} \) for accurate results.

Answer 1

The Correct Option is B

Step 1: Finding the amplitude of \(y_1\) Given: \[ y_1 = 5 \left[ \sin 2\pi t + \sqrt{3} \cos 2\pi t \right] \] Using the identity: \[ A = \sqrt{a^2 + b^2} \] where \( a = 5 \) and \( b = 5\sqrt{3} \). \[ A_1 = \sqrt{ (5)^2 + (5\sqrt{3})^2 } = \sqrt{ 25 + 75 } = \sqrt{100} = 10 \]
Step 2: Finding the amplitude of \(y_2\) Since \( y_2 = 5 \sin \left( 2\pi t + \frac{\pi}{4} \right) \), the amplitude is directly \( 5 \).
Step 3: Ratio of amplitudes \[ \frac{A_1}{A_2} = \frac{10}{5} = 2:1 \]

Answer 2

Step 1: Calculating the Amplitude of the First Wave \( y_1 \)

The given wave equation is:
\[ y_1 = 5 \left[ \sin(2\pi t) + \sqrt{3} \cos(2\pi t) \right] \]
To express this as a single sinusoidal wave, we use the identity:
\[ A = \sqrt{a^2 + b^2} \]
where \( a = 5 \) and \( b = 5\sqrt{3} \) (since the coefficient of \( \cos(2\pi t) \) is \( 5 \times \sqrt{3} \)).

\[ A_1 = \sqrt{5^2 + (5\sqrt{3})^2} = \sqrt{25 + 75} = \sqrt{100} = 10 \]
Step 2: Determining the Amplitude of the Second Wave \( y_2 \)

The second wave is given as:
\[ y_2 = 5 \sin\left(2\pi t + \frac{\pi}{4}\right) \]
Since this is already in the standard sinusoidal form, the amplitude is simply:
\[ A_2 = 5 \]
Step 3: Computing the Ratio of Amplitudes

The required ratio is:
\[ \frac{A_1}{A_2} = \frac{10}{5} = 2:1 \]
Final Answer:
\[ \boxed{2:1} \]