The displacement of a S.H.O. is given by the equation x=A cos{ωt+(π/ 8)}. At what time will it a

step 1 Hello and welcome to this video solution of numerate. So it is given that the displacement of a

The displacement of a S.H.O. is given by the equation x=A...

The displacement of a S.H.O. is given by the equation $\mathrm{x}=\mathrm{A}$ $\cos \{\omega t+(\pi / 8)\}$. At what time will it attain maximum velocity?
(A) $(3 \pi / 8 \omega)$
(B) $(8 \pi / 3 \omega)$
(C) $(3 \pi / 16 \omega)$
(D) $(\pi / 16 \pi)$.

Key Concepts

Phase Constant

The phase constant in the displacement equation represents the shift of the oscillatory motion in time. It adjusts the starting point of the cycle, ensuring that the function accurately describes the initial conditions of the motion. In problems involving maximum velocity, the phase constant plays a significant role in determining when the velocity peaks.

Angular Frequency

Angular frequency is a measure of how quickly the oscillatory cycle is completed and is denoted by the symbol ?. It relates the period of oscillation to the time and is used in the arguments of sine and cosine functions to represent the time evolution of the system.

Relationship Between Displacement and Velocity in SHM

In simple harmonic motion, the velocity of the oscillating object is the time derivative of the displacement. This leads to a sinusoidal velocity function that is phase-shifted relative to the displacement. Maximum velocity is achieved when the displacement is zero and the object is passing through the equilibrium point.

Simple Harmonic Motion (SHM)

Simple harmonic motion describes oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. It is fundamental in describing physical systems that exhibit periodic behavior, such as pendulums and springs, and is characterized mathematically by sine and cosine functions.

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