Question

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

Answer 1

If a particle that is said to move in a to and fro in a fixed motion under the action of the restoring force that is directly proportional to the fixed position that is displaced known as the simple harmonic motion. Consider the general equation of the simple harmonic motion to solve the given question.
Formula used:
The general equation of the SHM. That is,
$y = A\sin \omega t + B\cos \omega t$
Where,
$y$is the displacement,
$\omega$is the angular frequency of the motion,
$A,B$ are amplitudes,
$t$is the time

Complete step by step answer:
In the question, it is given that the initial phase of the particle that executes the simple harmonic motion. The initial phase is nothing but the time of the vibrating particle that corresponds to the time that is equal to zero.
Consider the general equation of the SHM. That is,
$y = A\sin \omega t + B\cos \omega t$
Where,
$y$is the displacement,
$\omega$is the angular frequency of the motion,
$A,B$ are amplitudes,
$t$is the time
Compare the general equation of SHM with the equation $y = 4\sin \omega t + 3\cos \omega t$
We have 4 in the place of A and 3 in the place of B.
The time value is zero as it is determined in the initial phase.
Therefore,
$\Rightarrow \tan \phi = \dfrac{3}{4}$
$\Rightarrow \phi = {\tan ^{ - 1}}\dfrac{3}{4}$
$\Rightarrow \phi = {36.9^ \circ }$
The value of $\phi$is approximately equal to the ${37^ \circ }$.
Therefore, the initial phase of the particle executing SHM with $y = 4\sin \omega t + 3\cos \omega t$is${37^ \circ }$
One of the initial phases is ${37^ \circ }$then another phase is found by subtracting ${37^ \circ }$with ${90^ \circ }$. That is,
$\Rightarrow \left( {{{90}^ \circ } - {{37}^ \circ }} \right)$
$\Rightarrow {53^ \circ }$

Hence option A and B is the correct answer.

Note: The simple harmonic motion can be represented in both sines and cosines. The total energy is conserved in the SHM. Every oscillatory motion is an SHM but every SHM is not an oscillatory motion. The simple harmonic motion has an angle produced in the phase point.