Two identical discs of same radius R are rotating about their axes
Problem:
Two identical discs of same radius R are rotating about their axes in opposite directions with the constant angular speed $\omega$. The discs are in the same horizontal plane. At time t=0, the point P and Q are facing each other as shown in the figure. The relative speed between the two points P and Q is $v_r$. In one time period (T) of rotation of the discs, $v_r$ as a function of time is best represented by
(IIT JEE 2012)




Solution:
The relative velocity of the point P w.r.t. the point Q is given by
\begin{align}
\label{pca:eqn:1}
\vec{v}_r=\vec{v}_P\vec{v}_Q.
\end{align}
It is easy to see that $\vec{v}_P=\vec{v}_Q=\omega R$ and angle traversed in time $t$ is $\omega t$. Thus, velocities of P and Q are
\begin{align}
&\vec{v}_P=\omega R(\sin\omega t\,\hat{\imath}\cos\omega t\,\hat{\jmath}), \nonumber\\
&\vec{v}_Q=\omega R(\sin\omega t\,\hat\imath\cos\omega t\,\hat\jmath). \nonumber
\end{align}
Substitute $\vec{v}_P$ and $\vec{v}_Q$ in above equation to get $\vec{v}_r=2\omega R\sin\omega t\,\hat\imath$ and thus $v_r=2\omega R\sin\omega t$. We encourage you to draw $\vec{v}_{P}$ and $\vec{v}_{Q}$ at time $t=0$, $T/4$, $T/2$, and $3T/4$ and see the values of $v_r$ at these instants.
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