The displacement for a particle performing S.H.M. is given by x=A cos(ωt+θ). If the initial posi

step 1 Hello and welcome to this video solution of numerate. The expression of the displacement of a pa

The displacement for a particle performing S.H.M. is given...

The displacement for a particle performing S.H.M. is given by $\mathrm{x}=\mathrm{A} \cos (\omega \mathrm{t}+\theta)$. If the initial position of the particle is $1 \mathrm{~cm}$ and its initial velocity is $\pi \mathrm{cms}^{-1}$, then what will be its initial phase ? The angular frequency of the particle is $\pi \mathrm{s}^{-1}$. (A) $(2 \pi / 4)$ (B) $(7 \pi / 4)$ (C) $(5 \pi / 4)$ (D) $(3 \pi / 4)$

The displacement for a particle performing S.H.M. is given by $\mathrm{x}=\mathrm{A} \cos (\omega \mathrm{t}+\theta)$. If the initial position of the particle is $1 \mathrm{~cm}$ and its initial velocity is $\pi \mathrm{cms}^{-1}$, then what will be its initial phase ? The angular frequency of the particle is $\pi \mathrm{s}^{-1}$.
(A) $(2 \pi / 4)$
(B) $(7 \pi / 4)$
(C) $(5 \pi / 4)$
(D) $(3 \pi / 4)$

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Key Concepts

Simple Harmonic Motion (SHM)

SHM describes the oscillatory motion of a particle where the restoring force is directly proportional to the negative of its displacement. The general equations involve sinusoidal functions that represent the displacement, velocity, and acceleration of the particle over time. This concept is fundamental in understanding periodic behavior in both mechanical and electrical systems.

Initial Conditions in SHM

The initial conditions, such as the initial displacement and velocity, are crucial in completely determining the motion of a particle in SHM. By substituting time t = 0 into the displacement and velocity equations, one obtains equations that relate the amplitude and phase constant of the oscillation to the known starting conditions.

Phase Angle and Its Calculation

The phase angle in SHM encapsulates the initial state of the oscillation. It is determined by the relationships cos(?) = (initial displacement)/A and sin(?) = -(initial velocity)/(A?). Calculating the phase angle often involves using inverse trigonometric functions and considering the correct quadrant based on the signs of the sine and cosine values provided by the initial conditions.

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