Question

Two simple harmonic motions are represented as \(y_{1}=10\,\sin \omega t\) and \(y_{2}=5\,\sin \omega t+5\,\cos \omega t\), both in cm. The ratio of the amplitudes of \(y_{2}\) (\(A_{2}\)) to that of \(y_{1}\) (\(A_{1}\)), is

• A. \(4:1\)
• B. \(\sqrt{2}:1\)
• C. \(1:1\)
• D. \(1:\sqrt{2}\)

Hint:

Combine the sine and cosine terms of y2 into a single sine using the resultant amplitude formula.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
A sum \(a\sin\omega t+b\cos\omega t\) is one SHM of amplitude \(\sqrt{a^{2}+b^{2}}\).

Step 2: Amplitudes
\(A_1=10\text{ cm}\)
\(A_2=\sqrt{5^{2}+5^{2}}=5\sqrt{2}\text{ cm}\)

Step 3: Calculation
\(\dfrac{A_2}{A_1}=\dfrac{5\sqrt{2}}{10}=\dfrac{1}{\sqrt{2}}\)

Step 4: Compare the options
The ratio \(1:\sqrt{2}\) is the last option.

Final Answer:
The ratio of the amplitudes A2 to A1 is 1 : root 2. \[ \boxed{\text{(D) }1:\sqrt{2}} \]