Question

Two particles $A$ and $B$ are moving in uniform circular motion in concentric circles of radii ${{r}_{A}}$ and ${{r}_{B}}$ with speed ${{V}_{A}}$ and ${{V}_{B}}$ respectively. Their time-period of rotation is the same. The ratio of angular speed of $A$ to that of $B$ will be:
$\begin{aligned} & A.\,\,{{r}_{A}}:{{r}_{B}} \\\ & B.\,\,{{V}_{A}}:{{V}_{B}} \\\ & C.\,\,{{r}_{B}}:{{r}_{A}} \\\ & D.\,\,1:1 \\\ \end{aligned}$

Answer 1

It is given in the question that time-period is the same for both the particles, this is the most important point that is needed to solve the problem. The radius and speed of both the particles are mentioned just to provide some extra details regarding the particle. Find relationship between angular speed and time-period to solve the problem.
Formula used:
$Angular\,Speed\,(\omega )=\dfrac{2\pi }{Time-period(T)}$

Complete answer:
According to the question, following are the details given related to the particles $A$ and $B$:
Two particles $A$ and $B$ are moving in uniform circular motion in concentric circles.
${{r}_{A}}$, ${{r}_{B}}$ are the radius of the two particles $A$ and $B$.
${{V}_{A}},{{V}_{B}}$are velocity of the two particles $A$ and $B$.
${{T}_{A}}={{T}_{B}}$are the time-period of the two particles $A$ and $B$, which are equal.

Now,
The formula for angular speed for the two particles $A$ and $B$ can be written as:
${{\omega }_{A}}=\dfrac{2\pi }{{{T}_{A}}}$ and ${{\omega }_{B}}=\dfrac{2\pi }{{{T}_{B}}}$
Hence,
$\dfrac{{{\omega }_{A}}}{{{\omega }_{B}}}=\dfrac{{}^{2\pi }/{}_{{{T}_{A}}}}{{}^{2\pi }/{}_{{{T}_{B}}}}=\dfrac{{{T}_{B}}}{{{T}_{A}}}$
Since, ${{T}_{A}}={{T}_{B}}$
Then, $\dfrac{{{\omega }_{A}}}{{{\omega }_{B}}}=\dfrac{1}{1}$
$\Rightarrow {{\omega }_{A}}:{{\omega }_{B}}=1:1$.

Therefore, the correct answer is Option (D).

Note:
The radius and velocity of the two particles are given in the question, to provide extra details about the particles. If you know the formula of angular speed, then you can understand that radius and velocity have nothing to do when trying to get a solution.