Two particles $A$ and $B$ are moving in a uniform circular motion in concentric circles of radii $r_A$ and $r_

Question

Two particles $A$ and $B$ are moving in a uniform circular motion in concentric circles of radii $r_A$ and $r_B$ with speeds $v_A$ and $v_B$ respectively. Their time periods of rotation are the same. The ratio of the angular speed of $A$ to that of $B$ will be:

Accepted Answer

1. **Definition of Angular Speed:** The angular speed $\omega$ of a particle performing uniform circular motion is related to its time period $T$ (time for one complete revolution) by the formula $\omega = \frac{2\pi}{T}$ .
2. **Analyze the Condition:** The problem states that the time periods of rotation for both particles are the same, i.e., $T_A = T_B$.
3. **Calculate Ratio:**
   $$ \omega_A = \frac{2\pi}{T_A} $$
   $$ \omega_B = \frac{2\pi}{T_B} $$
   Since $T_A = T_B$, it implies that $\omega_A = \omega_B$.
4. **Conclusion:** The ratio of their angular speeds is $\omega_A : \omega_B = 1 : 1$. The radii and linear speeds are not needed to determine this specific ratio given the constant time period condition.

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