The speed of a particle executing motion changes with time...

The speed of a particle executing motion changes with time according to the equation $\mathrm{y}=\mathrm{a} \sin \omega \mathrm{t}+\mathrm{b} \cos \omega \mathrm{t}$, then $\ldots \ldots \ldots$ (A) Motion is periodic but not a S.H.M. (B) It is a S.H.M. with amplitude equal to $\mathrm{a}+\mathrm{b}$ (C) It is a S.H.M. with amplitude equal to $\mathrm{a}^{2}+\mathrm{b}^{2}$

The speed of a particle executing motion changes with time according to the equation $\mathrm{y}=\mathrm{a} \sin \omega \mathrm{t}+\mathrm{b} \cos \omega \mathrm{t}$, then $\ldots \ldots \ldots$
(A) Motion is periodic but not a S.H.M.
(B) It is a S.H.M. with amplitude equal to $\mathrm{a}+\mathrm{b}$
(C) It is a S.H.M. with amplitude equal to $\mathrm{a}^{2}+\mathrm{b}^{2}$

Key Concepts

Phase Transformation

Phase transformation is the process of rewriting a linear combination of sinusoidal functions into a single sinusoid by combining their amplitude and phase shift. This method simplifies the analysis of oscillatory systems by presenting the motion in a more compact and interpretable form.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of oscillatory motion where the restoring force is proportional to the displacement from an equilibrium position. This property results in periodic motion that can be modeled by sine or cosine functions, leading to a predictable and repetitive behavior such as that seen in springs and pendulums.

Sinusoidal Functions

Sinusoidal functions, including sine and cosine functions, are used to model periodic phenomena because of their inherent cyclical pattern. In the context of oscillatory motion, these functions capture the time-dependent behavior of the system and are fundamental in describing waveforms and oscillations.

Amplitude Calculation

Amplitude is a measure of the maximum displacement from the equilibrium position in periodic motion. When a motion is expressed as a sum of sine and cosine components, trigonometric identities can be used to convert this sum into a single sinusoidal function with an amplitude defined as the square root of the sum of the squares of the coefficients.

Periodic Motion

Periodic motion refers to any motion that repeats itself at regular intervals or periods. It is characterized by identifiable parameters such as period, frequency, and amplitude, and encompasses a wide range of phenomena including both simple harmonic motion and more complex cyclic behaviors.

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