$A$ tunnel is dug along the diameter of the Earth of mass $M$. $A$ mass $m$ is released at a distance $X$ from the center,and the time period of oscillation is $T$. If a mass $4m$ is released from the same point,what will be the time period of oscillation?

$Assertion$ : The length of the day is slowly increasing.
$Reason$ : The dominant effect causing a slowdown in the rotation of the earth is the gravitational pull of other planets in the solar system.

Select the correct choice$(s)$:

$A$ solid sphere of radius $R$ gravitationally attracts a particle of mass $m$ placed at a distance of $3R$ from its centre with a force $F_{1}$. Now,a spherical cavity of radius $\frac{R}{2}$ is made in the sphere (as shown in the figure) such that the cavity is tangent to the centre of the sphere and the outer surface. The force on the particle becomes $F_{2}$. The value of $F_{1} : F_{2}$ is:

$A$ uniform sphere $A$ with radius $R$ exerts a gravitational force $F$ on a small particle $B$ situated at a distance $2R$ from the centre of the sphere. $A$ spherical portion of diameter $R$ is cut from the sphere $A$ such that its centre lies on the line connecting the centre of sphere $A$ and particle $B$. If $F^{\prime}$ is the new gravitational force between the remaining part of the sphere $A$ and the particle $B$, then find the correct relation between $F$ and $F^{\prime}$.

$A$ hemispherical shell of mass $2M$ and radius $6R$ and a point mass $M$ are performing circular motion due to their mutual gravitational interaction. Their positions are shown in the figure at any moment of time during motion. If $r_1$ and $r_2$ are the radii of the circular paths of the hemispherical shell and the point mass respectively,and $\omega_1$ and $\omega_2$ are the angular speeds of the hemispherical shell and the point mass respectively,then choose the correct option.