Question

Initial phase of the particle executing SHM with ${\text{y = 4sin}}\omega {\text{t + 3cos}}\omega {\text{t}}$ is:
A) ${53^ \circ }$
B) ${37^ \circ }$
C) \[\;{90^ \circ }\]
D) ${45^ \circ }$

Answer 1

Hint: Simple Harmonic Motion: A particle is said to be in simple harmonic motion if it moves to and fro about a fixed position under the action of restoring force which is directly proportional to the displacement from the fixed position and is always directed toward the fixed position.
Initial phase in Simple harmonic motion: The phase of a vibrating particle corresponding to time t=0 is called the initial phase or epoch.

Complete step-by-step solution:
The general representation of simple harmonic motion is,
${\text{y = Asin}}\omega {\text{t + Bcos}}\omega {\text{t}}$, where A and B are the amplitude of the motion, $\omega $= angular frequency of the motion,
$t$= time period, $Y$= Displacement of the simple harmonic motion

Given details: ${\text{y = 4sin}}\omega {\text{t + 3cos}}\omega {\text{t}}$, here \[A = 4\], $B = 3$
Initial phase is determined when the time period, t=0.
Therefore,
${\text{tan }}\phi {\text{ = }}\dfrac{3}{4}$
$ \rightarrow \phi = {\text{ta}}{{\text{n}}^{ - 1}}(\dfrac{3}{4})$
\[ = {36.9^ \circ } \approx {37^ \circ }\]
If one initial phase is \[{37^ \circ }\] then the other initial phase will be ${(90 - 37)^0} = {53^0}$

The correct options are (A) and (B).

Note: Properties of Simple harmonic motion:
-The simple harmonic motion can be represented in sine and cosine.
-The total energy of the particle executing simple harmonic motion is conserved.
-Examples of simple harmonic motion: swing, pendulum, bungee jumping, cradle, etc.
-Every oscillatory motion is a period in motion but all periodic motions are not oscillatory motion.
-Oscillatory motion is to and fro motion about the mean position.
-The phase of a point in simple harmonic motion is the angle produced by the point, in a uniform circular motion whose projection is that simple harmonic motion, with the initial point of motion at the center of the circular motion.