$A$ particle of mass $m$ moves in a circular orbit under the central potential field,$U(r) = -\frac{C}{r}$,where $C$ is a positive constant. The correct radius-velocity graph of the particle's motion is:

If a person with a spring balance and a body hanging from it goes up and up in an aeroplane,then the reading of the weight of the body as indicated by the spring balance will

$A$ solid sphere of uniform density and radius $4$ units is located with its centre at the origin $O$ of coordinates. Two spheres of equal radii $1$ unit with their centres at $A(-2, 0, 0)$ and $B(2, 0, 0)$ respectively are taken out of the solid leaving behind spherical cavities as shown in the figure.

$A$ 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.

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?

$A$ spherically symmetric gravitational system of particles has a mass density $\rho = \begin{cases} \rho_0 & \text{for } r \leq R \\ 0 & \text{for } r > R \end{cases}$ 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: