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I am working on two different SHMs: $$P=a\sin\omega t \\ Q=a\sin(\omega t +\phi)$$ where $\omega$ = angular velocity, $\phi$ = phase constant, $P,Q$ = displacement at a instant, $a$ = amplitude

Now due to phase constant the mean position is changed.

Question: Does the positions ($Q$) where the maximum acceleration, zero acceleration, max velocity and zero velocity occur for the first time also change with respect to $P$?

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The only difference between the $P=a\sin\omega t $ and the $Q=a\sin(\omega t +\phi)$ description is that you are observing the same motion but starting the clock, setting $t=0$, at difference parts of the motion.

So when $t=0$ you have $P=0$ and $Q=a\sin \phi$ and you might say that motion $Q$ is phase $\phi$ ahead of motion $P$ but the mean position over a period is zero for both descriptions of the motion.

The same argument can be followed through for velocity and acceleration.

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If you have an object undergoing simple harmonic motion with a position $x(t)$ given by $$x(t) = A \sin (\omega t + \phi),$$

then the velocity of the object is: $$v(t) = \frac{\text{d}x}{\text{d}t} = A \omega \,\cos(\omega t + \phi),$$

and the acceleration is: $$a(t) = \frac{\text{d}v}{\text{d}t} = - A \omega^2 \, \sin(\omega t + \phi),$$

as it should be.

In your case, you have two different "position" functions, one for the oscillator $P$ and one for the oscillator $Q$. You can use the above formulae to calculate the quantities you are asked to calculate, and find if they are the same, or different.

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