Question

The frequency of a particle performing SHM is $12Hz$. Its amplitude is $4cm$. Its initial displacement is $2cm$ towards a positive extreme position. Its equation for displacement is
A $x = 0.04\cos \left( {24\pi t + \dfrac{\pi }{6}} \right)$
B $x = 0.04\sin \left( {24\pi t} \right)$
C $x = 0.04\sin \left( {24\pi t + \dfrac{\pi }{6}} \right)$
D $x = 0.04\cos \left( {24\pi t} \right)$

Answer 1

Simple Harmonic Motion or SHM is a motion in which the restoring force is directly proportional to the displacement of the object from its starting position. The direction of this restoring force is always towards the starting position. The acceleration of a particle is in simple harmonic motion
The frequency of oscillation is the number of oscillations performed by the object in one second. When the object oscillates in SHM, the position of the object from its mean position is the initial displacement.
Formula used:
The formula of displacement of the particle is
$x = A\sin \left( {\omega t + \phi } \right)$

Complete answer:
Let, the frequency of particle $f = 12Hz$
As we know the angular frequency $\omega = 2\pi f$or, $\omega = 24\pi rad/s$
Amplitude is $A = 4cm$
From the equation $x = A\sin \left( {\omega t + \phi } \right)$ we get,
$\Rightarrow x = 4\sin \left( {24\pi t + \phi } \right)$
Initial displacement $x = 0$ at $t = 0$
$\therefore 2 = 4\sin \left( {24\pi \times 0 + \phi } \right)$
$\Rightarrow \sin \phi = \dfrac{2}{4} = \dfrac{1}{2}$
$\therefore \phi = {30^ \circ } = \dfrac{\pi }{6}$

Hence the correct answer is option C.

Note:
There are two types of simple harmonic motion. A) linear motion, B) angular motion
In linear motion an object moves to and from a fixed point along with a straight line. Where restoring force or acceleration of the moving object is proportional to its displacement. But, in Angular motion, an object oscillates with angular displacement from a fixed point. Where resting torque or angular acceleration of the moving object is proportional to its angular displacement.
In the mean position force acting on the object is zero. From the mean position maximum displacement is presented as amplitude in simple harmonic motion.