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Two particles move on a circular path (one just inside and the other just outside) with angular velocities $\omega$ and $5\omega$ starting from the same point. Then
they cross each other at points on the path subtending $90^o$ at the centre if their angular velocities are in the same sense.
they cross each other at points on the path subtending an angle of $60^o$ at the centre if their angular velocities are oppositely directed.
they cross at intervals of time $\frac{\pi }{{3\omega }}$ if their angular velocities are oppositely directed.
All of the above
Solution
case $1$
When both move in opposite direction
$\theta_{1}=w t$
$\theta_{2}=5 w t$
They meet each other when $\theta_{2}+\theta_{1}=2 \pi$
$\therefore 5 w t+w t=2 \pi$
$\therefore w t=\frac{\pi}{3}=60^{\circ}$
le. the bodies cross each other at points subtending an angle of $60^{\circ}$ if their angular velocities are directed opposite to each other.
case $2$
When both move in same direction $\theta_{1}=w t$
$\theta_{2}=5 w t$
They meet each other when $\theta_{2}-\theta_{1}=2 \pi$
$\therefore 5 w t-w t=2 \pi$
$\therefore w t=\frac{\pi}{2}=90^{\circ}$
i.e. the bodies cross each other at points subtending an angle of $90^{\circ}$ if their angular velocities are similar.
Now, when opposity directed, beat frequency$:$
$=\boldsymbol{n}_{2}-\boldsymbol{n}_{1}$
$=\frac{5 w}{2 \pi}-\left(\frac{-w}{2 \pi}\right)$
$=\frac{3 w}{\pi}$
$\therefore T=\frac{\pi}{3 w}$
