Question

When two displacements represented by $ y_1 = a \sin \, (\omega \, t)$ and $ y_2 = b \, cos \, ( \omega \, t )$ are superimposed the motion is

• A. simple harmonic with amplitude $ \sqrt{ a^2 + b^2 }$
• B. simple harmonic with amplitude $ \frac{ (a + b)}{ 2} $
• C. not a simple harmonic
• D. simple harmonic with amplitude $ \frac{ a}{ b} $

Answer 1

The Correct Option is A

To solve this problem, we need to understand the concept of superposition of harmonic motions. Here, we have two displacements:

• The first displacement, $y_1 = a \sin (\omega t)$, is a simple harmonic motion along the sine axis.
• The second displacement, $y_2 = b \cos (\omega t)$, is a simple harmonic motion along the cosine axis.

We are asked to find the resultant motion when these two displacements are superimposed. The resultant displacement can be expressed as:

$y = y_1 + y_2 = a \sin (\omega t) + b \cos (\omega t)$

This expression can be rewritten as a single sinusoidal function using the identity for phase addition:

$y = R \sin(\omega t + \phi)$

where $R$ is the amplitude of the resultant wave and $\phi$ is the phase angle. The amplitude $R$ is given by:

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

Hence, the resultant motion is simple harmonic with an amplitude of $ \sqrt{ a^2 + b^2 }$.

Conclusion:

The correct answer is: simple harmonic with amplitude $ \sqrt{ a^2 + b^2 }$.

Explanation for Other Options:

• $ \frac{(a+b)}{2}$: This suggests an average value-like interpretation, which is not correct for the resultant amplitude of two perpendicular harmonic motions.
• Not simple harmonic: The combined motion remains simple harmonic as long as it can be described by a single sinusoidal function, which it is.
• $ \frac{a}{b}$: This could represent a ratio-related interpretation but does not apply to determining the resultant amplitude.