Plot the corresponding reference circle for each of the following…

Question

Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial $(t=0)$ position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: ( $x$ is in cm and t is in s ).
a. $x=-2 \sin \left(3 t+\frac{\pi}{3}\right)$
b. $x=\cos \left(\frac{\pi}{6}-t\right)$
c. $x=3 \sin \left(2 \pi t+\frac{\pi}{4}\right)$
d. $x=2 \cos \pi t$

✓

Answer

a. $x =2 \cos \left(3 t+\frac{\pi}{3}+\frac{\pi}{2}\right)$
Radius of the reference circle, $r =$ amplitude of $SHM =2 cm$,
At $t =0, x =-2 \sin \frac{\pi}{3}=\frac{-2 \sqrt{3}}{2}=-\sqrt{3} cm$
Also $\omega t =3 t \therefore \omega=3 rad / s$
$\cos \phi_0=-\frac{\sqrt{3}}{2}, \phi_0=150^{\circ}$
The reference circle is, thus, as plotted below.

b. $x=\cos \left(t-\frac{\pi}{6}\right)$
Radius oif circle, $r =$ amplitude of $SHM =1 cm$.
At $t =0, x =\cos \frac{\pi}{6}=\frac{\sqrt{3}}{2} cm$
Also $\omega t =1 t \Rightarrow \omega=1 rad / s$
$
\cos \phi_0=\frac{\sqrt{3}}{2}, \quad \phi_0=-\frac{\pi}{6}
$
The reference circle is, thus as plotted below.

c. $x =3 \cos \left(2 \pi t+\frac{\pi}{4}+\frac{\pi}{2}\right)$
Here, radius of reference circle, $r =3 cm$ and at $t =0, x =3 \sin \frac{\pi}{4}=\frac{\sqrt{3}}{2} cm$
$\omega t =2 \pi t \Rightarrow \omega=2 \pi rad / s$
$
\cos \phi_0=\frac{\frac{\sqrt{3}}{2}}{3}=-\frac{1}{\sqrt{2}}
$
Therefore, the reference circle is being shown below.

d. $x =2 \cos \pi t$
Radius of reference circle, $r =2 cm$ and at $t =0, x =2 cm$
$
\begin{aligned}
& \therefore \omega t=\pi t \text { or } \omega=\pi rad / s \\
& \cos \phi_0=1, \phi_0=0
\end{aligned}
$
The reference circle is plotted below.

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