Three identical particles $A, B$,and $C$ of mass $100 \, kg$ each are placed in a straight line with $AB = BC = 13 \, m$. The gravitational force on a fourth particle $P$ of the same mass is $F$,when placed at a distance $13 \, m$ from the particle $B$ on the perpendicular bisector of the line $AC$. The value of $F$ will be approximately $....G$.

Mean solar day is the time interval between two successive noons when the sun passes through the zenith point (meridian). $A$ sidereal day is the time interval between two successive transits of a distant star through the zenith point (meridian). By drawing an appropriate diagram showing the earth's spin and orbital motion,show that the mean solar day is $4\,\text{min}$ longer than the sidereal day. In other words,distant stars would rise $4\,\text{min}$ early every successive day.

Take the mean distance of the moon and the sun from the earth to be $0.4 \times 10^6 \, km$ and $150 \times 10^6 \, km$ respectively. Their masses are $8 \times 10^{22} \, kg$ and $2 \times 10^{30} \, kg$ respectively. The radius of the earth is $6400 \, km$. Let $\Delta F_1$ be the difference in the forces exerted by the moon at the nearest and farthest points on the earth and $\Delta F_2$ be the difference in the force exerted by the sun at the nearest and farthest points on the earth. Then,the number closest to $\frac{\Delta F_1}{\Delta F_2}$ is

Which one of the following plots represents the variation of gravitational field on a particle with distance $r$ due to a thin spherical shell of radius $R$? ($r$ is measured from the centre of the spherical shell)

If the height of a satellite from the surface of the Earth is increased,then its

$A$ particle of mass $M$ is situated at the centre of a spherical shell of same mass and radius $a.$ The magnitude of the gravitational potential at a point situated at $a/2$ distance from the centre,will be