Two bodies of mass $10 ext{ kg}$ and $5 ext{ kg}$ moving in concentric orbits of radii $R$ and $r$ such that

Question

Two bodies of mass $10	ext{ kg}$ and $5	ext{ kg}$ moving in concentric orbits of radii $R$ and $r$ such that their periods are the same. Then the ratio between their centripetal acceleration is:

Accepted Answer

1.  **Formula for Centripetal Acceleration:** The centripetal acceleration ($a_c$) of a body moving in a circular path of radius $R$ with angular speed $\omega$ is given by $a_c = \omega^2 R$ .
2.  **Angular Speed:** The angular speed is related to the time period ($T$) by $\omega = \frac{2\pi}{T}$ .
3.  **Application:** Since both bodies have the same time period ($T$), they must have the same angular speed ($\omega$).
    *   For the first body: $a_1 = \omega^2 R$
    *   For the second body: $a_2 = \omega^2 r$
4.  **Ratio:** Taking the ratio of their accelerations:
    $\frac{a_1}{a_2} = \frac{\omega^2 R}{\omega^2 r} = \frac{R}{r}$.
    Note that the centripetal acceleration depends only on the radius and angular speed (or period), and is independent of the mass of the body.

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