A thin uniform annular disc (see figure) of mass M has outer radius 4R and inner radius 3R. The work required to take a unit mass from point P on its axis to infinity is

Column II shows five systems in which two objects are labelled as X and Y. Also in each case a point P is shown. Column I gives some statements about X and/or Y. Match these statements to the appropriate system(s) from Column II:

Column I | Column II | ||
---|---|---|---|

(A) | The force exerted by X on Y has a magnitude $$Mg$$. | (P) | Block Y of mass M left on a fixed inclined plane X, slides on it with a constant velocity. |

(B) | The gravitational potential energy of X is continuously increasing. | (Q) | Two rings magnets Y and Z, each of mass M, are kept in frictionless vertical plastic stand so that they repel each other. Y rests on the base X and Z hangs in air in equilibrium. P is the topmost point of the stand on the common axis of the two rings. The whole system is in a lift that is going up with a constant velocity. |

(C) | Mechanical energy of the system X + Y is continuously decreasing. | (R) | A pulley Y of mass $$m_0$$ is fixed to a table through a clamp X. A block of mass M hangs from a string that goes over the pulley and is fixed at point P of the table. The whole system is kept in a lift that is going down with a constant velocity. |

(D) | The torque of the weight of Y about point is zero. | (S) | A sphere Y of mass M is put in a non-viscous liquid X kept in a container at rest. The sphere is released and it moves down in the liquid. |

(T) | A sphere Y of mass M is falling with its terminal velocity in a viscous liquid X kept in a container. |

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