If the equation for displacement of two particles executing...

If the equation for displacement of two particles executing S.H.M. is given by $\mathrm{y}_{1}=2 \sin (10 \mathrm{t}+\theta)$ and $\mathrm{y}_{2}=3 \cos 10 \mathrm{t}$ respectively, then the phase difference between the velocity of two particles will be $\ldots \ldots \ldots$ (A) $-\theta$ (B) $\theta$ (C) $\theta-(\pi / 2)$ (D) $\theta+(\pi / 2)$.

If the equation for displacement of two particles executing S.H.M. is given by $\mathrm{y}_{1}=2 \sin (10 \mathrm{t}+\theta)$ and $\mathrm{y}_{2}=3 \cos 10 \mathrm{t}$
respectively, then the phase difference between the velocity of two particles will be $\ldots \ldots \ldots$
(A) $-\theta$
(B) $\theta$
(C) $\theta-(\pi / 2)$
(D) $\theta+(\pi / 2)$.

Key Concepts

Phase Difference in Oscillatory Motion

Phase difference quantifies the angular separation between the phases of two periodic functions. It is a measure of how much one waveform is shifted in time relative to another, and is critical when comparing oscillatory behaviors, especially when they are expressed with different trigonometric functions or include additional phase constants.

Differentiation of Sinusoidal Functions

Differentiating a sinusoidal function, whether sine or cosine, involves applying the chain rule. This process not only scales the amplitude by the angular frequency but also results in a phase shift. In the context of oscillatory motion, this property is used to transform displacement equations into velocity equations, illustrating the inherent ?/2 phase difference.

Simple Harmonic Motion

SHM is a type of periodic motion where the restoring force on an object is proportional to its displacement from equilibrium and acts in the opposite direction. It is commonly modeled using sine and cosine functions, and understanding SHM involves analyzing amplitude, frequency, and phase, which describe the oscillatory behavior.

Relationship between Displacement and Velocity in SHM

In simple harmonic motion, velocity is the time derivative of displacement. This differentiation introduces a phase shift of ?/2 radians relative to the displacement. Recognizing this shift is essential for comparing the phase relationships of different oscillatory quantities such as displacement and velocity.

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