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?

For a satellite to orbit around the earth,which of the following must be true?

$A$ 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:

State whether the following statements are true or false:
$(a)$ The value of acceleration due to gravity increases as we go to a depth $d$ below the Earth's surface.
$(b)$ If the escape velocity for an object of mass $m$ on the Earth's surface is $11.2 \, km/s$,then the escape velocity for an object of mass $2m$ is $22.4 \, km/s$.
$(c)$ If the Earth's gravitational attraction suddenly disappears,the mass of an object becomes zero,but its weight remains the same.

The gravitational force acting on a particle,due to a solid sphere of uniform density and radius $R$,at a distance of $3 R$ from the centre of the sphere is $F_1$. $A$ spherical hole of radius $(R / 2)$ is now made in the sphere as shown in the figure. The sphere with hole now exerts a force $F_2$ on the same particle. The ratio of $F_1$ and $F_2$ is

$A$ small ball of mass $m$ is released at a height $R$ above the earth's surface,as shown in the figure. The ball enters a narrow groove and reaches a maximum depth of $R/2$ inside the earth before coming to rest momentarily. The groove contains an ideal spring of spring constant $K$ and natural length $R$. Find the value of $K$ if $R$ is the radius of the earth and $M$ is the mass of the earth.