The earth takes $24\; h$ to rotate once about its axis. How much time (in $min$) does the sun take to shift by $1^o$ when viewed from the earth?

The figure shows the variation of the gravitational acceleration $a_g$ of four planets with the radial distance $r$ from the center of the planet for $r \ge R$ (where $R$ is the radius of the planet). Plots $1$ and $2$ coincide for $r \ge R_2$,and plots $3$ and $4$ coincide for $r \ge R_4$. The sequence of the planets in the descending order of their densities is:

Particles of masses $m, 2m, 3m$ and $4m$ are placed at the corners of a square of side $a$. What is the net gravitational force on a particle of mass $m$ placed at the center?

Suppose the acceleration due to gravity at the earth’s surface is $g \text{ m/s}^2$ and at the surface of the moon it is $g' \text{ m/s}^2$. An $M \text{ kg}$ passenger travels from the earth to the moon in a spaceship moving with a constant velocity. Neglecting all other celestial objects, which curve best represents the net gravitational force on the passenger as a function of time?

Match List-$I$ with List-$II$:

List-$I$	List-$II$
$(A)$ Kinetic energy of planet	$(1)$ $-\frac{GMm}{a}$
$(B)$ Gravitational potential energy of Sun-planet system	$(2)$ $\frac{GMm}{2a}$
$(C)$ Total mechanical energy of planet	$(3)$ $\frac{GM}{r}$
$(D)$ Escape energy at the surface of planet for unit mass object	$(4)$ $-\frac{GMm}{2a}$

(Where $a=$ radius of planet orbit,$r=$ radius of planet,$M=$ mass of Sun,$m=$ mass of planet)
Choose the correct answer from the options given below:

$A$ satellite of mass $m$ is at a distance $a$ from a star of mass $M$. The speed of the satellite is $u$. Suppose the law of universal gravity is $F = -G \frac{Mm}{r^{2.1}}$ instead of $F = -G \frac{Mm}{r^2}$. Find the speed of the satellite when it is at a distance $b$ from the star.