Gravitation · Physics · JEE Advanced

A particle of mass m, and angular momentum ℓ is moving in a circular orbit of radius r₀ under the influence of an attractive force $\vec{F}(r)=-\frac{k}{r^2} \hat{r}$. Keeping its angular momentum unchanged, the particle is displaced radially by a small distance $\delta r \ll r_0$, due to which its radial distance varies periodically. The corresponding time period is :

Consider a star of mass m₂ kg revolving in a circular orbit around another star of mass m₁ kg with m₁ \gg m₂. The heavier star slowly acquires mass from the lighter star at a constant rate of $\gamma$ kg/s. In this transfer process, there is no other loss of mass. If the separation between the centers of the stars is r, then its relative rate of change $\frac{1}{r}\frac{dr}{dt}$ (in s⁻¹) is given by:

A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M(\gg m)$. The particle is moving in a circular orbit of radius $r_0$ with time period $T_0$ around the mass $M$. Then, the particle is subjected to an additional central force, corresponding to the potential energy $V_{\mathrm{c}}(r)=m \alpha / r^3$, where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_0$ in the combined gravitational potential due to $M$ and $V_{\mathrm{c}}(r)$, but with a new time period $T_1$, then $\left(T_1^2-T_0^2\right) / T_1^2$ is given by

[G is the gravitational constant.]

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

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.

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

Two spherical stars $A$ and $B$ have densities $\rho_{A}$ and $\rho_{B}$, respectively. $A$ and $B$ have the same radius, and their masses $M_{A}$ and $M_{B}$ are related by $M_{B}=2 M_{A}$. Due to an interaction process, star $A$ loses some of its mass, so that its radius is halved, while its spherical shape is retained, and its density remains $\rho_{A}$. The entire mass lost by $A$ is deposited as a thick spherical shell on $B$ with the density of the shell being $\rho_{A}$. If $v_{A}$ and $v_{B}$ are the escape velocities from $A$ and $B$ after the interaction process, the ratio $\frac{v_{B}}{v_{A}}=\sqrt{\frac{10 n}{15^{1 / 3}}}$. The value of $n$ is __________ .