Question

The amplitude and the periodic time of a SHM are 5cm and 6s respectively. At a distance of 2.5cm away from the mean position, the phase will be
A. $\dfrac{\pi}{3}$
B. $\dfrac{\pi}{4}$
C. $\dfrac{\pi}{6}$
D. $\dfrac{5\pi}{12}$

Answer 1

SHM (Simple Harmonic Motion) is the type of periodic motion in which the net restoring force F acting on the body is proportional to the displacement x from the equilibrium position and is directed opposite to the displacement, i.e., towards the equilibrium point. The body performing SHM is known as a simple harmonic oscillator (SHO).

Complete step by step answer:
The motion of a particle moving along a straight line with an acceleration whose direction is always towards a fixed point on the line and whose magnitude is proportional to the distance from the fixed point is called simple harmonic motion. In mechanics and physics, simple harmonic motion (sometimes abbreviated SHM) is a special type of periodic motion where the restoring force on the moving object is directly proportional to the object's displacement magnitude and acts towards the object's equilibrium position.

It results in an oscillation which, if uninhibited by friction or any other dissipation of energy, continues indefinitely.The displacement of SHM can be represented by a sine or cosine function such as, $x = a\sin \omega t$ , where, “a” is the amplitude and $\omega$ is the angular frequency.
$x = a\sin \omega t$
$\Rightarrow 2.5 = 5\sin \omega t$
$\Rightarrow {\text{Let, }}\omega t{\text{ = }}\phi$
Therefore,
$2.5 = 5\sin \phi$
Now find the value of $\sin \phi$
$\Rightarrow \sin \phi = \dfrac{{2.5}}{5} = \dfrac{1}{2}$
Get the value of $\phi$ by finding the value of sine inverse
$\Rightarrow \phi = {\sin ^{ - 1}}\left( {\dfrac{1}{2}} \right)$
$\therefore \phi = \dfrac{\pi }{6}$

Hence, the correct answer is option C.

Additional information: The acceleration developed in the motion due to the restoring force is directly proportional to its displacement from the equilibrium position. The Force or acceleration is always directed opposite to the displacement i.e., towards the mean position. The displacement can be represented by a sine or cosine function such as, $x = a\sin \omega t$ where, a is the amplitude and $\omega$ is the angular frequency. The velocity of the body is maximum at the centre and minimum at extreme position.

Note: The students should know all the values of standard inverse trigonometry. Here in this question, we have substituted $\omega t = \phi$ as the phase is equal to $\omega t$ so substituting it to $\phi$ we get an answer easily. Here $\phi = \dfrac{{2\pi t}}{6}$ and $\omega t = \dfrac{{2\pi t}}{6}$. There are two types of SHM, Linear SHM: Example, the vertical oscillations of a loaded spring suspended from a rigid support. Motion of needle of sewing machine. And Angular SHM: Example, Motion of pendulum.