Question

Two particles are moving in different circles in the same plane with different angular velocities as shown in the figure. At $t=0$, initial positions of particles $A$ and $B$ are shown by dots on the respective circles. The initial distance between particles is $1m$. Particle $A$ move anticlockwise in the first circle whereas $B$ moves clockwise in the second circle. Angle described (rotated) by $A$ and $B$ in time $t$ are ${\theta }_A=\left(\frac{\pi }{2}t\right)$ and ${\theta }_B=\left(\pi t\right)$ respectively.. Here $\theta$ is in radian and $t$ is in second. The radius of each circle as shown in the diagram. Determine the time after which the distance between A and B is 1m again?

Answer 1

Time after A and B is 1m again when both are at same initial position.

$T_1=\frac{2\pi }{{\omega }_1}=\frac{2\pi }{\pi /2}=4sec$

$T_2=\frac{2\pi }{{\omega }_2}=\frac{2\pi }{\pi }=2sec$

$\Delta T=T_1-T_2=2sec$

So after 2 revolution of particle which time period $T_2$ and one revolution which particle of time period $T_1$

So time taken to for 1m distance $T_1=2T_2=4sec$