A spherically symmetric gravitational system of particles has a mass density
$$\rho = \left\{ {\matrix{ {{\rho _0}} & {for} & {r \le R} \cr 0 & {for} & {r > R} \cr } } \right.$$
Where $$\rho_0$$ is a constant. A test mass can undergo circular motion under the influence of the gravitational field of particles. Its speed V as a function of distance $$r(0 < r < \infty)$$ from the centre of the system is represented by
STATEMENT - 1
An astronaut in an orbiting space station above the Earth experiences weightlessness.
and
STATEMENT - 2
An object moving around the Earth under the influence of Earth's gravitational force is in a state of 'free-fall'.
Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II and indicate your answer by darkening appropriate bubbles in the 4 $$\times$$ 4 matrix given in the ORS.
| Column I | Column II | ||
|---|---|---|---|
| (A) | GM$$_e$$M$$_s$$ G - universal gravitational constant, M$$_e$$ - mass of the earth, M$$_s$$ - mass of the Sun |
(P) | (volt) (coulomb) (metre) |
| (B) | $${{3RT} \over M}$$ R - universal gas constant, T - absolute temperature, M - molar mass |
(Q) | (kilogram) (metre)$$^3$$ (second)$$^{-2}$$ |
| (C) | $${{{F^2}} \over {{q^2}{B^2}}}$$ F - force, q - charge, B - magnetic field |
(R) | (metre)$$^2$$ (second)$$^{-2}$$ |
| (D) | $${{G{M_e}} \over {{R_e}}}$$ G - universal gravitational constant, M$$_e$$ - mass of the earth R$$_e$$ - radius of the earth |
(S) | (farad) (volt)$$^2$$ (kg)$$^{-1}$$ |
A system of binary stars of masses $m_{\mathrm{A}}$ and $m_{\mathrm{B}}$ are moving in circular orbits of radii $r_{\mathrm{A}}$ and $r_R$, respectively. If $\mathrm{T}_A$ and $\mathrm{T}_B$ are the time periods of masses $m_A$ and $m_B$ respectively, then
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