Question

A particle executes SHM with periodic time of $ 6 $ seconds. The time taken for traversing a distance of half the amplitude from the mean position is:
(A) $ 3\sec $
(B) $ 2\sec $
(C) $ 1\sec $
(D) $ 1/2\sec $

Answer 1

Hint: Simple Harmonic Motion in Physics is a type of motion periodic motion that is linear in nature. Its motion is always directed towards some fixed point in its path. To solve this equation we have to use the equation of motion of Simple Harmonic Motion.

Formula used:
 $ y = A\sin (\omega t) $
where, $ y $ is the instantaneous displacement of the particle,
 $ A $ is the maximum displacement or the amplitude of the particle executing SHM,
 $ \omega $ is the angular frequency of the particle,
and $ t $ is the instantaneous time of the particle.
 $ \omega = \dfrac{{2\pi }}{T} $
where, $ T $ is the time period of the particle.

Complete step by step answer:
We know that the equation of motion of the particle executing SHM is given by
 $ y = A\sin (\omega t) $
where, $ y $ is the instantaneous displacement of the particle,
 $ A $ is the maximum displacement or the amplitude of the particle executing SHM,
 $ \omega $ is the angular frequency of the particle,
and $ t $ is the instantaneous time of the particle.
In the question, it is given that instantaneous displacement is half the amplitude of the particle.
Therefore we can write that $ y = \dfrac{A}{2} $ .
Hence, $ \dfrac{A}{2} = A\sin (\omega t) $
Cancelling $ A $ from both sides we get,
 $ \dfrac{1}{2} = \sin (\omega t) $
We know that the value of $ \sin \dfrac{\pi }{6} = \dfrac{1}{2} $
Hence we can say, $ \omega t = \dfrac{\pi }{6} $
Now, we know the formula,
 $ \omega = \dfrac{{2\pi }}{T} $
where, $ T $ is the time period of the particle.
Therefore, $ \dfrac{{2\pi t}}{T} = \dfrac{\pi }{6} $
Cancelling $ \pi $ from both sides we get,
 $ \dfrac{{2t}}{T} = \dfrac{1}{6} $
 $ \Rightarrow t = \dfrac{T}{{12}} $
From the question we have $ T = 6\sec $
Substituting these values in the above equation we get,
 $ t = \dfrac{6}{{12}} $
or, $ t = \dfrac{1}{2}\sec $
Hence, option (D) is the correct answer.

Note:
The force acting on the particle executing Simple Harmonic Motion is proportional to the displacement, i.e., the distance from the fixed point. The function of the force is to bring back the body to its equilibrium position and hence the name restoring force.