Explain the concepts of a reference particle and a reference circle,and show that simple harmonic motion $(SHM)$ is the projection of uniform circular motion on a diameter of the reference circle.

(N/A) At time $t=0$,the particle is at position $P_{1}$ and its position vector $\overrightarrow{OP}_{1}$ makes an angle $\phi$ with the positive $X$-axis.
The projection of $OP_{1}$ on the $X$-axis is $OP_{1}^{\prime}$.
At time $t=t$,the particle has undergone an angular displacement $\omega t$,reaching point $P_{2}$,and its position vector $\overrightarrow{OP}_{2}$ makes an angle $\omega t+\phi$ with the $X$-axis.
The projection of the position vector $OP_{2}$ on the $X$-axis is $OP_{2}^{\prime}$.
As the particle $P$ moves on a circle,its perpendicular projection on the $X$-axis is given by $x(t)=A \cos(\omega t+\phi)$. This represents the $X$-component of the position vector at any time.
This equation is the general equation of $SHM$.
From this,it is concluded that the projection of uniform circular motion on a diameter of the reference circle is $SHM$.
The particle moving on a uniform circular path is called the reference particle,and the circular path of the reference particle is called the reference circle.
If the projection of the reference particle is taken on the $Y$-axis,the displacement of the particle on the $Y$-axis is $y(t)=A \sin(\omega t+\phi)$.