Two particles $A$ and $B$ are moving in a same plane anti clock wise on two different concentric circles with different constant angular velocities $2 ω$ and $ω$ respectively. At t=0 , positions of the particles are as shown in the diagram. After time $t = \frac{π}{ω}$ . Find

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position of $B$ w.r.t $A$ is $- 3 r \hat{i}$

velocity of $B$ w.r.t $A$ is $- 4 ω r \hat{j}$

angular velocity of $B$ w.r.t $A$ is $\frac{4 ω}{3}$

acceleration of $B$ w.r.t $A$ is $6 ω^{2} r \hat{i}$

answer is A, B, C, D.

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Detailed Solution

$θ_{A} = 2 ω \frac{π}{ω} = 2 p, v_{A} = 2 ω r \hat{j} θ_{B} = ω \frac{π}{ω} = π, v_{B} = 2 ωr (- \hat{j}) {\vec{v}}_{B / A} = {\vec{v}}_{B} - {\vec{v}}_{A} = 2 ωr (- \hat{j}) - 2 ωr (\hat{j}) = - 4 ωr \hat{j} {\vec{r}}_{B / A} = {\vec{r}}_{B} - {\vec{r}}_{A} = 2 r (- \hat{i}) - (r) \hat{i} = - 3 r \hat{i} a_{B / A} = ω^{2} . 2 r \hat{i} - {(2 ω)}^{2} r (- \hat{i}) = 6 ω^{2} r \hat{i} ω_{B / A} = \frac{v_{B / A}}{r_{B / A}} = \frac{4 ω}{3}$