A particle starts SHM at time t=0 . Its amplitude is A and...

A particle starts SHM at time $t=0 .$ Its amplitude is $A$ and angular frequency is $\omega .$ At time $t=0$, its kinetic energy is $\frac{E}{4}$, where $E$ is total energy. Assuming potential energy to be zero at mean position, the displacement-time equation of the particle can be written as (A) $x=A \cos \left(\omega t+\frac{\pi}{6}\right)$ (B) $x=A \sin \left(\omega t+\frac{\pi}{3}\right)$ (C) $x=A \sin \left(\omega t-\frac{2 \pi}{3}\right)$ (D) $x=A \cos \left(\omega t-\frac{\pi}{6}\right)$

A particle starts SHM at time $t=0 .$ Its amplitude is $A$ and angular frequency is $\omega .$ At time $t=0$, its kinetic energy is $\frac{E}{4}$, where $E$ is total energy. Assuming potential energy to be zero at mean position, the displacement-time equation of the particle can be written as
(A) $x=A \cos \left(\omega t+\frac{\pi}{6}\right)$
(B) $x=A \sin \left(\omega t+\frac{\pi}{3}\right)$
(C) $x=A \sin \left(\omega t-\frac{2 \pi}{3}\right)$
(D) $x=A \cos \left(\omega t-\frac{\pi}{6}\right)$

Key Concepts

Mechanical Energy in SHM

In the context of SHM, mechanical energy is conserved and is continuously exchanged between kinetic and potential forms. The total energy, a function of the amplitude and angular frequency, helps in determining the state of motion at various instants, including the initial distribution between kinetic and potential energy.

Angular Frequency

Angular frequency (?) is a measure of how rapidly the oscillations occur, expressed in radians per second. In SHM, it connects time to displacement and is inversely related to the period of the oscillation.

Phase Constant

The phase constant is a parameter in the sinusoidal function that shifts the waveform along the time axis. It is crucial for incorporating initial conditions, such as initial displacement and velocity, ensuring that the mathematical description correctly represents the state of the oscillating system at t = 0.

Simple Harmonic Motion (SHM)

Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to displacement from an equilibrium position. This results in oscillatory behavior typically described by sine or cosine functions, with the motion exhibiting predictable and repetitive patterns.

Amplitude

Amplitude is the maximum displacement from the equilibrium position in oscillatory motion. It reflects the energy involved in the motion and determines the 'size' of the oscillations in a simple harmonic system.

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