Assertion: Two SHMs are given by y1=10 sin(3 πt+(π)/(4)) and y2=5 sin(3 πt)+√(3) cos(3 πt) Reaso

step 1 Hello students in this question we have given the assertion which is having two SHM waves that i

Assertion: Two SHMs are given by y1=10 sin(3 πt+(π)/(4)) and...

Assertion: Two SHMs are given by $y_{1}=10 \sin \left(3 \pi t+\frac{\pi}{4}\right)$ and $y_{2}=5 \sin (3 \pi t)+\sqrt{3} \cos$ $(3 \pi t)$ Reason: $y_{2}$ represents two SHMs each of amplitude 5 and so total amplitude is 10 , same as that of $y_{1}$. (A) A (B) $\mathrm{B}$ (C) $\mathrm{C}$ (D) D

Assertion: Two SHMs are given by $y_{1}=10 \sin \left(3 \pi t+\frac{\pi}{4}\right)$ and $y_{2}=5 \sin (3 \pi t)+\sqrt{3} \cos$
$(3 \pi t)$
Reason: $y_{2}$ represents two SHMs each of amplitude 5 and so total amplitude is 10 , same as that of $y_{1}$.
(A) A
(B) $\mathrm{B}$
(C) $\mathrm{C}$
(D) D

Key Concepts

Simple Harmonic Motion (SHM)

SHM refers to a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the direction opposite to that displacement. This type of motion is typically described using sine or cosine functions, which capture the oscillatory behavior of the system over time.

Sinusoidal Function Representation

Sinusoidal functions, such as sine and cosine, are used to mathematically model periodic phenomena like SHM. These functions can incorporate amplitude, angular frequency, and phase shift, where the phase shift accounts for any horizontal displacement in time, allowing for a complete description of the oscillatory behavior.

Vector Addition of Sinusoids

When two sinusoids of the same frequency but different phases are added, their combined effect is determined by vector addition. Each sinusoid can be thought of as a rotating vector (phasor), and the resultant amplitude is the magnitude of the vector sum. This approach requires the use of trigonometric identities or methods like the cosine or sine addition formulas.

Resultant Amplitude Calculation

The resultant amplitude of a sum of sinusoids is calculated using the square root of the sum of the squares of the individual amplitudes (when the sinusoids are orthogonal in phase) or by considering both amplitude and phase differences through vector addition. It is important to note that amplitudes do not simply add arithmetically unless the sinusoids are exactly in phase.

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