Points to Ponder

The period $T$ is the shortest time after which the motion exactly repeats itself. Because of this, the motion also repeats after any integer multiple $nT$, where $n = 1, 2, 3, \dots$

Not every motion that repeats in time is simple harmonic. For motion to be simple harmonic, it must obey the force law $F = -k x$, where the restoring force is directly proportional to the displacement and opposite in direction.

A particle can move in a circle under two different kinds of forces: an inverse-square law (like gravity for planets) or a simple harmonic force in two dimensions given by $\mathbf{F} = -m\omega^2 \mathbf{r}$. In the SHM case, uniform circular motion occurs only when the oscillations along the $x$ and $y$ axes have a phase difference of exactly $\pi/2$. For instance, a particle starting at $(0, A)$ with velocity $(\omega A, 0)$ under the force $-m\omega^2\mathbf{r}$ will trace a circle of radius $A$.

For linear SHM with a fixed angular frequency $\omega$, exactly two initial conditions are needed to completely determine the motion. These can be any of the following pairs: (i) initial position and initial velocity, (ii) amplitude and phase constant, or (iii) total energy and phase constant.

Because of point 4, if you know the amplitude (or the total energy), the phase constant is fixed by either the initial position or the initial velocity — you don't need both.

Adding two SHMs with arbitrary amplitudes and phases does not always give a periodic result. The sum is periodic only when the ratio of their frequencies is a rational number (one frequency is an integer multiple of the other). However, any periodic motion can be broken down into an infinite sum of harmonic motions with the right amplitudes (Fourier series).

The time period of SHM is independent of amplitude, energy, and phase constant. This is different from planetary orbits, where Kepler's third law shows that the period depends on the semi-major axis (and hence on the energy).

A simple pendulum's motion is only approximately simple harmonic when the angular displacement is small. …