The displacement of a particle varies with time as x=12 sinωt-16 sin^3 ωt (in .cm) . If its moti

Step 1:u000AWe are given the displacement of a particle as $xu003D12 u005Csin u005Comega tu002D16 u005Csin ^{3} u005Comega t

The displacement of a particle varies with time as x=12...

The displacement of a particle varies with time as $x=12 \sin \omega t-16 \sin ^{3} \omega t$ (in $\left.\mathrm{cm}\right) .$ If its motion is
SHM, then its maximum acceleration is :
(a) $12 \omega^{2}$
(b) $36 \omega^{2}$
(c) $144 \omega^{2}$
(d) $\sqrt{192} \omega^{2}$

Key Concepts

Simple Harmonic Motion

Simple harmonic motion is a type of oscillatory motion where the restoring force is directly proportional to the displacement, resulting in a sinusoidal time dependence. It is characterized by a constant period, frequency, and amplitude, and is a fundamental concept in the study of oscillatory systems in physics.

Kinematics in SHM

In simple harmonic motion, the displacement, velocity, and acceleration of a particle are interrelated. The acceleration is obtained as the second derivative of the displacement function with respect to time, and it is directly proportional to the displacement but in the opposite direction, following the relation a(t) = -?² x(t).

Maximum Acceleration in SHM

The maximum acceleration in an oscillatory system undergoing simple harmonic motion occurs when the displacement reaches its maximum value, or amplitude. It is given by the formula |a_max| = ?²A, where A is the amplitude and ? is the angular frequency, reflecting the system's sensitivity to changes in displacement.

Trigonometric Transformations

Trigonometric identities can be used to simplify complex oscillatory functions by converting sums or powers of sine and cosine functions into a standard sinusoidal form. This process allows one to identify the effective amplitude and phase, thereby making it easier to analyze properties like maximum acceleration in the context of simple harmonic motion.

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