Question

The phase difference between the displacement and acceleration of particles executing S.H.M. radian is:
(A) $\dfrac{\pi }{4}$
(B) $\dfrac{\pi }{2}$
(C) $\pi $
(D) $2\pi $

Answer 1

Hint: To solve this question we must know that whilst simple harmonic motion is a simplification, it is still a very good approximation. Simple harmonic motion is important in research to model oscillations for example in wind turbines and vibrations in car suspensions. Simple Harmonic Motion or SHM is defined as a motion in which the restoring force is directly proportional to the displacement of the body from its mean position. The acceleration of a particle executing simple harmonic motion is given by, a $(\mathrm{t})=-\omega^{2} \mathrm{x}(\mathrm{t}) .$ Here, $\omega$ is the angular velocity of the particle. The word displacement implies that an object has moved, or has been displaced. Displacement is defined to be the change in position of an object.

Complete step by step answer
It is known that in mechanics and physics, simple harmonic motion 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.
We know that the expression for the simple harmonic motion is given as:$\mathrm{va} \sin (\omega \mathrm{t}+\phi)$
Now differentiating we get: $\mathrm{v}=\mathrm{a} \omega \cos (\omega \mathrm{t}+\phi)$
Further, differentiating again, $\mathbf{a}=-\mathbf{a} \omega^{2} \sin (\omega+\phi)$
Now taking minus sign $\mathrm{a}=\mathrm{a} \omega^{2} \sin (\omega \mathrm{t}+\phi+\pi)$
So, phase difference between displacement and acceleration will be:-
$=\omega \mathrm{t}+\phi+\pi-\omega \mathrm{t}+\phi$
$=\omega \mathrm{t}+\phi+\pi-\omega \mathrm{t}-\phi$
$=\pi$

Hence, we can say that the correct option is option C.

Note: We know that the Simple Harmonic motion is periodic in nature that is it repeated its position again and again, and in harmonic motion Hooke's Law is applicable which states that Force is always proportional to displacement but due to air resistance and other factors the amplitude goes on decreasing which restricts the motion to be Simple Harmonic Motion. Graph of displacement against time in simple harmonic motion. where F is force, x is displacement, and k is a positive constant. This is exactly the same as Hooke's Law, which states that the force F on an object at the end of a spring equals \[-kx\], where k is the spring constant.
Thus, we can conclude that therefore, every oscillatory motion is periodic but all periodic motions are not oscillatory. Furthermore, simple harmonic motion is the simplest type of oscillatory motion. This motion takes place when the restoring force acting on the system is directly proportional to its displacement from its equilibrium position.