Simple Harmonic Motion: Relation between angular motion and linear to and from motion What is angular frequency in simple Harmonic Motion?
If simple harmonic motion is a linear to and fro motion then whose angular frequency are we talking about? A linearly moving body cannot have an angular frequency right? I know that we compare it with uniform circular motion, but then how can we associate angular quantities for linear translation? Is the frequency associated to the air particle/surrounding particles that disturbed due to that SHM?
And why was the motion compared to UCM only?
 A: Simple harmonic motion is not just to and fro motion - it is to and fro motion where the displacement from the central position over time follows a sine wave. There are other types of to and fro motion which are not simple harmonic motion - for example, where the displacement from the central position over time follows a triangle wave rather than a sine wave.
The connection with uniform circular motion is that simple harmonic motion is the projection on the x-axis of the position of an object moving with uniform circular motion in the x-y plane. If the object is moving in a circle of radius $R$ with frequency $f$ then at time $t$ its position is
$\left(R\sin (2\pi f t), R\cos (2\pi f t)\right)$
so the projection of its position on the x-axis is $R \sin(2\pi f t)$. A particle moving with simple harmonic motion $x(t)=R \sin(2\pi f t)$ is said to be moving with frequency $f$ because it takes $\frac 1 f$ seconds to move from one extreme point $x=R$ to the opposite extreme point $x=-R$ and back again.
A: Simple harmonic motion is a clockwise rotation about the unit circle in the scaled phase space coordinates $(x/x_{xmax},\quad v/x_{max}\omega)$.   If the oscillator is a mass $m$ on a spring constant $k$, then $\omega=\sqrt{\frac{k}{m}}$.
$$
\begin{align}
x &=x_{max}\quad\cos(\omega t) \\
v=\frac {dx}{dt} &=-x_{max}\omega\quad sin(\omega t) \\
\end {align}
$$
Notice that
$$
 (\frac{x}{x_{max}})^2 + (\frac{v}{x_{max}\omega})^2=1 \\
$$
$$
\begin{bmatrix}
x/x_{max}\\
v/x_{max}\omega\\ 
\end{bmatrix}_t=
\begin{bmatrix}
cos(\omega t) & sin(\omega t) \\
-sin(\omega t) & cos(\omega t)  \\
\end{bmatrix}
\begin{bmatrix}
1\\
0\\  
\end{bmatrix}_{t=0}
$$
