Question

Two bodies of mass $10\,kg$ and $5\,kg$ moving in concentric orbits of radii R and r such that their periods are the same. Then the ratio between their centripetal acceleration is
A. $\dfrac{R}{r} \\ $
B. $\dfrac{r}{R} \\ $
C. $\dfrac{{{R^2}}}{{{r^2}}} \\ $
D. $\dfrac{{{r^2}}}{{{R^2}}}$

Answer 1

Hint:In order to solve this question, we will use the general formula of centripetal acceleration for the circular motion and using this we will solve for the ratio of the centripetal acceleration of two bodies.

Formula used:
For a circular motion, centripetal acceleration is given by,
$a = \dfrac{{{v^2}}}{r}$
where v is velocity and r is the radius of a circular path.
Time period is given by,
$T = \dfrac{{2\pi r}}{v}$

Complete step by step solution:
According to the question, let us suppose velocities of given two masses $10kg$ and $5kg$ are ${v_1},{v_2}$ and their radius of circular path given to us R and r, given that time period are equal so,
$T = \dfrac{{2\pi R}}{{{v_1}}}$ and $T = \dfrac{{2\pi r}}{{{v_2}}}$
So, we have
$\dfrac{{2\pi R}}{{{v_1}}} = \dfrac{{2\pi r}}{{{v_2}}} \\
\Rightarrow \dfrac{{{v_1}}}{{{v_2}}} = \dfrac{R}{r} \to (i) \\ $
Now, the centripetal acceleration of both bodies are,
${a_1} = \dfrac{{{v_1}^2}}{R} \\
\Rightarrow {a_2} = \dfrac{{{v_2}^2}}{r} \\ $
On dividing both values of centripetal acceleration, we get
$\dfrac{{{a_1}}}{{{a_2}}} = {\left( {\dfrac{{{v_1}}}{{{v_2}}}} \right)^2}\dfrac{r}{R}$
Using value from equation (i) we get
$\dfrac{{{a_1}}}{{{a_2}}} = {\left( {\dfrac{R}{r}} \right)^2}\dfrac{r}{R} \\
\therefore \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{R}{r} \\ $
So, the ratio of the centripetal acceleration of both the bodies is $\dfrac{R}{r}$.

Hence, the correct answer is option A.

Note: It should be remembered that, centripetal acceleration and time period of a body moving in circular motion is independent of the mass of the body as centripetal acceleration and time period depends only upon the velocity of the body and radius of the circular path. and concentric orbits are those when two bodies move in circular orbits having different radii but their centers are the same.