The periodic time of a S.H.O. oscillating about a fixed point is 2 s. After what time will the

step 1 Hello and welcome to this video solution of numerate. So here we are given that the periodic tim

The periodic time of a S.H.O. oscillating about a fixed...

The periodic time of a S.H.O. oscillating about a fixed point is $2 \mathrm{~s}$. After what time will the kinetic energy of the oscillator become $25 \%$ of its total energy?
(A) $1 / 12 \mathrm{~s}$
(B) $1 / 6 \mathrm{~s}$
(C) $1 / 4 \mathrm{~s}$
(D) $1 / 3 \mathrm{~s}$.

Key Concepts

Simple Harmonic Motion

Simple Harmonic Motion (SHM) describes a type of periodic oscillation where the restoring force is directly proportional to the displacement from the equilibrium position. This concept is fundamental to understanding systems like springs and pendulums, and involves characteristics such as amplitude, period, and phase.

Angular Frequency and Time

Angular frequency (?) relates to the period (T) of oscillation by the formula ? = 2?/T. This parameter is crucial in determining how the oscillatory motion evolves over time and is used to convert between the time domain and the cyclic properties of the oscillation.

Kinetic Energy in SHM

In simple harmonic motion, the kinetic energy of the oscillator varies sinusoidally with time and is often expressed as a fraction of the total mechanical energy. When the oscillator is released from maximum displacement, the kinetic energy increases from zero and reaches a maximum when the displacement is zero. Understanding this energy partitioning is key for analyzing the dynamics of the system.

Sinusoidal Energy Variation

The time dependence of the kinetic energy in a harmonic oscillator can be described using sinusoidal functions, typically as a squared sine or cosine function. This mathematical representation connects the phase of the oscillation to the distribution of kinetic and potential energies throughout the cycle.

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