Question

A particle executes SHM with a time period of $4s$. Find the time taken by the particle to go particle to go directly from its mean position to half of its amplitude.?

Answer 1

We are given that a particle is executing SHM. It SHM the extent to which the body is displaced from its mean position is depending on the restoring force acting on it i.e., restoring force is directly proportional to the displacement and this restoring force always acts towards the mean position.
Amplitude can be defined as the distance from the mean position to any of the extreme positions.

Complete Step by Step Solution:
We are assuming that the particle is starting its journey from its mean position, then
$X = A\sin \omega t$
Where A is the amplitude
Omega is the angular velocity
t is the time in seconds.
Since the particle is moving from mean position to half of the amplitude
$\Rightarrow X = \dfrac{A}{2}$
$\Rightarrow \dfrac{A}{2} = A\sin \omega t$
$\Rightarrow \omega t = \dfrac{\pi }{6}$
We know that,$\omega = \dfrac{{2\pi }}{T}$
$\Rightarrow T = 4s$
$\Rightarrow \omega = \dfrac{{2\pi }}{4}$
$\Rightarrow \omega = \dfrac{\pi }{2}rad{\sec ^{ - 1}}$
We got the value of omega now we can find the time taken.
$\Rightarrow \dfrac{\pi }{2} \times t = \dfrac{\pi }{6}$
$\Rightarrow t = \dfrac{1}{3}s$

Additional Information:
Oscillation is the periodic variation of matter between two values or about its central value and vibration can be defined as the mechanical oscillations of an object.
Damped oscillations are produced through processes of controlling vibrations such as mechanical vibrations through the emission of energy. These types of oscillations fade away with time.
In undamped oscillations restoring force is proportional to displacement. Here the oscillation never fades and the magnitude of oscillation remains the same. An example is the case of Alternating current.

Note:
Damped oscillations are classified again as underdamped oscillations, critically damped oscillations, overdamped oscillations. These are classified on the basis of damping constant.
Under damped oscillations, the damping constant is less than one and the oscillations reach the stable state very slowly.
Critically damped oscillations, the damping constant is one.
Over damped oscillations, the damping constant is greater than one and they reach the equilibrium point very slowly.