Question

How do you find the center of motion of a particle moving in simple harmonic motion of period $ 8\;hours $ and amplitude $ 6\;m $ , when $ t=3hours $ and $ x=4m $ ?

Answer 1

Hint :The center of the motion is a point where the velocity is maximum. The center of the motion can also be defined as the initial phase. Hence, we need to find the equation of the SHM and obtain the value of the initial phase or center of motion from it.

Complete Step By Step Answer:
Let us note down the given data;
Time period of the SHM $ T=8h $
Amplitude of the SHM $ A=6m $
Time at which given displacement is found $ t=3h $
Displacement of the particle at the given moment $ x=4m $
Initial phase of the SHM $ \phi =? $
Now, the center of the motion is the reference point for the SHM. Every quantity is measured with respect to the center of the motion.
Center of motion can also be defined as the point from which the motion is initiated. Hence, its value is equal to the value of the initial phase.
Also, the center of motion can be said as the lowest point of SHM, as both the extreme positions are equidistant from the center.
Now, at the lowest point the velocity is maximum. Hence velocity is maximum at the center of the motion.
Now, we know that the angular velocity is calculated as,
 $ \omega =\dfrac{2\pi }{T} $
Substituting the value of time period,
 $ \omega =\dfrac{2\pi }{8h} $
 $ \therefore \omega =0.785rad{{h}^{-1}} $
We know that the general equation for SHM is given as,
 $ x=A\sin \left( \omega t+\phi \right) $
Substituting the specific values for the given case,
 $ \therefore x=6\sin \left( 0.785t+\phi \right) $
Now, we are given a condition for time and displacement. Substituting the condition,
 $ \therefore 4=6\sin \left( 0.785\times 3h+\phi \right) $
 $ \therefore \dfrac{4}{6}=\sin \left( 2.356+\phi \right) $
Applying the inverse function on both sides,
 $ \therefore 2.356+\phi ={{\sin }^{-1}}\left( \dfrac{2}{3} \right) $
 $ \therefore 2.356+\phi =0.73 $
Hence, the value of initial phase is,
 $ \therefore \phi =0.73-2.356 $
 $ \therefore \phi =-1.626 $
Hence, the center of the motion is $ \phi =-1.626 $

Note :
Here, we must have a clear understanding about the center of the motion. It is mostly the point from where the SHM initiates. All the measurements are taken with respect to the center of motion. Hence, their values depend on the center of motion.