Question

A particle executes Simple Harmonic Motion along a straight line so that its period is\[12seconds\] . Find the time it takes in travelling a distance equal to half of its amplitude from equilibrium position is:

Answer 1

Hint: An oscillatory motion where the acceleration of the particle at any position is directly proportional to the displacement from the mean position is called as Simple Harmonic Motion (S.H.M).
In this type of oscillatory motion displacement, velocity and acceleration and force vary with respect to time in a way that can be described by either sine or cosine functions called sinusoids.
Understanding the concepts of Simple Harmonic Motion is very useful and forms an important tool in understanding the characteristics of waves(sound, light) and alternating currents.

Formula used:
A S.H.M can be expressed as
\[y = a{\text{ }}sin\omega t\] or it can be \[y = a{\text{ }}\cos \omega t\]; Where \[a = \] amplitude of oscillation.

Complete step-by-step solution:
Time period \[\left( T \right) = 12{\text{ }}seconds\]
Distance travelled \[\left( y \right) = 0.5 \times a\]
We use \[y = asin\omega t\] to find phase angle (\[\omega t\]) by putting value of Distance travelled (y) in the equation
Therefore,\[0.5a = asin\omega t\]
\[ \Rightarrow 0.5 = sin\omega t\]
$\sin \dfrac{\pi }{6} = \sin \omega t$
 $\dfrac{\pi }{6} = \omega t$
Now, angular frequency, $\omega = 2\dfrac{\pi }{T}$
Hence $2\dfrac{\pi }{T} \times t = \dfrac{\pi }{6}$
$ \Rightarrow t\; = \;\dfrac{T}{{12}}sec$
$ \Rightarrow t\; = \;1sec$ [ since time period\[T = 12{\text{ }}seconds\]]
Required time taken is \[1sec\]

Note: For the motion of a particle to be simple harmonic the necessary and sufficient condition is that the restoring force must be proportional to displacement of the particle from equilibrium position. Simple Harmonic Motions are oscillatory and periodic but all oscillatory motions are not considered as SHM. Any sine or cosine function is oscillatory in nature because its value varies harmonically from +1 to -1. So, sine and cosine functions are called harmonic functions (or periodic functions). In case of S.H.M, the displacement is a harmonic function of time alone. The restoring force is proportional to the displacement from mean position and it is always directed towards the equilibrium (mean) position.