If the kinetic energy of a particle executing S.H.M. is given by K=K0 cos^2 ωt, then the displac

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If the kinetic energy of a particle executing S.H.M. is...

If the kinetic energy of a particle executing S.H.M. is given by $\mathrm{K}=\mathrm{K}_{0} \cos ^{2} \omega \mathrm{t}$, then the displacement of the particle is given by $\ldots \ldots$ (A) $\left\{\mathrm{K}_{0} / \mathrm{m} \omega^{2}\right\} \sin \omega \mathrm{t}$ (B) $\left\{\left(2 \mathrm{~K}_{0}\right) /\left(\mathrm{m} \omega^{2}\right)\right\}^{1 / 2} \sin \omega t$ (C) $\left\{2 \omega^{2} / \mathrm{mK}_{0}\right\} \sin \omega t$ (D) $\left\{2 \mathrm{~K}_{0} / \mathrm{m} \omega\right\}^{1 / 2} \sin \omega t$

If the kinetic energy of a particle executing S.H.M. is given by $\mathrm{K}=\mathrm{K}_{0} \cos ^{2} \omega \mathrm{t}$, then the displacement of the particle is given by $\ldots \ldots$
(A) $\left\{\mathrm{K}_{0} / \mathrm{m} \omega^{2}\right\} \sin \omega \mathrm{t}$
(B) $\left\{\left(2 \mathrm{~K}_{0}\right) /\left(\mathrm{m} \omega^{2}\right)\right\}^{1 / 2} \sin \omega t$
(C) $\left\{2 \omega^{2} / \mathrm{mK}_{0}\right\} \sin \omega t$
(D) $\left\{2 \mathrm{~K}_{0} / \mathrm{m} \omega\right\}^{1 / 2} \sin \omega t$

Key Concepts

Simple Harmonic Motion (SHM)

SHM is a type of periodic motion where the force acting on a particle is directly proportional to its displacement from the equilibrium position and acts in the opposite direction. This creates oscillatory motion that can be described mathematically using sine or cosine functions, with key parameters such as amplitude, angular frequency, and phase angle defining the motion.

Kinetic Energy in SHM

In SHM, the kinetic energy varies with time because the speed of the oscillating particle changes continuously. The kinetic energy is given by (1/2) m v^2, and when the displacement is expressed as a sine or cosine function, the velocity (and therefore kinetic energy) becomes a function of time, typically involving squared sine or cosine terms.

Amplitude Determination from Energy

The amplitude of oscillation in SHM can be linked to the maximum kinetic energy of the system. When kinetic energy is provided as a function of time, comparing it to the standard form (1/2) m A^2 ?^2 cos^2(?t + ?) allows one to extract the amplitude A by equating the maximum kinetic energy terms. This relationship is crucial for moving between energy descriptions and displacement equations.

Displacement Equation in SHM

The displacement in SHM is typically represented by a sine or cosine function that reflects the periodic nature of the motion. By integrating the relationship between the velocity and displacement, and using the determined amplitude, the displacement as a function of time can be expressed explicitly, showing how the position oscillates between plus and minus the amplitude with a period determined by the angular frequency.

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