Q.Draw graphs of displacement, velocity and acceleration against phase angle, for a particle performing linear S.H.M. from

[!TLDR] From mean position: x=sine, v=cosine, a=inverted sine. From positive extreme: x=cosine, v=inverted sine, a=inverted cosine. Conclusions: all periodic in T; phase(v)-phase(x)=pi/2; phase(a)-phase(v)=pi/2; phase(a)-phase(x)=pi. [!ANSWER] See the five listed conclusions.

(a) Starting from the mean position ($\phi=0$): $x=A\sin\omega t$ (sine curve: 0 at t=0, +A at T/4, 0 at T/2, -A at 3T/4, 0 at T), $v=A\omega\cos\omega t$ (cosine curve: $+A\omega$ at t=0, 0 at T/4, $-A\omega$ at T/2, 0 at 3T/4, $+A\omega$ at T), $a=-A\omega^2\sin\omega t$ (inverted-sine curve: 0 at t=0, $-A\omega^2$ at T/4, 0 at T/2, $+A\omega^2$ at 3T/4, 0 at T) -- see Fig. 5.6. (b) Starting from the positive extreme ($\phi=\pi/2$): $x=A\cos\omega t$ (cosine curve), $v=-A\omega\sin\omega t$ (inverted-sine curve), $a=-A\omega^2\cos\omega t$ (inverted-cosine curve) -- see Fig. 5.7; each curve here is simply the corresponding curve of case (a), shifted by a quarter period. Conclusions drawn from comparing the graphs: (1) x, v and a are all periodic functions of time, each of period T; (2) the displacement-time and acceleration-time curves are always of the same ('sine') type, while the velocity-time curve is of the other ('cosine') type; (3) there is a phase difference of $\pi/2$ radian between displacement and velocity; (4) there is a phase difference of $\pi/2$ radian between velocity and acceleration; (5) there is a phase difference of $\pi$ radian between displacement and acceleration (i.e. they are always of opposite sign, consistent with $a=-\omega^2x$); and the shapes of all curves repeat after a phase of $2\pi$, i.e. after time T. [!ANSWER] Graphs as described; the five conclusions listed above.

Substitute $\phi=0$ and $\phi=\pi/2$ into the general x(t), v(t), a(t) expressions, sketch each as a function of phase/time, and read off the mutual phase differences.

Missing the pi/2 phase-lead pattern (v ahead of x, a ahead of v) and instead only noting that all three curves 'look similar', without quantifying the phase differences between them.