Let the speed of the planet at the perihelion $P$ in figure shown below be $v_{_P}$ and the Sun-planet distance $\mathrm{SP}$ be $r_{_P}.$ Relation between $(r_{_P},~v_{_P})$ to the corresponding quantities at the aphelion $(r_{_A},~v_{_A})$ is:

Three equal masses of $m$ kg each are fixed at the vertices of an equilateral triangle $ABC.$ What is the force acting on a mass $2m$ placed at the centroid $G$ of the triangle? 
(Take $AG=BG=CG=1$ m.)

       

1. $Gm^2(\hat{i}+\hat{j})$
2. $Gm^2(\hat{i}-\hat{j})$
3. zero
4. $2Gm^2(\hat{i}+\hat{j})$

Three equal masses of $m$ kg each are fixed at the vertices of an equilateral triangle $ABC.$ What is the force acting on a mass $2m$ placed at the centroid $G$ of the triangle if the mass at the vertex $A$ is doubled?
Take $AG=BG=CG=1~\text{m}.$

1.   $Gm^{2}   \left(\hat{i} + \hat{j}\right)$
2. $Gm^{2}   \left(\hat{i} - \hat{j}\right)$
3. $0$
4. $2Gm^{2} \hat{j}$

The potential energy of a system of four particles placed at the vertices of a square of side $l$ (as shown in the figure below) and the potential at the centre of the square, respectively, are:

1. $- 5 . 41 \dfrac{Gm^{2}}{l}$ and $0$

2. $0$ and $- 5 . 41 \dfrac{Gm^{2}}{l}$ 

3. $- 5 . 41 \dfrac{Gm^{2}}{l}$ and $- 4 \sqrt{2} \dfrac{Gm}{l}$

4. $0$ and $0$

Two uniform solid spheres of equal radii ${R},$ but mass ${M}$ and $4M$ have a centre to centre separation $6R,$ as shown in the figure. The two spheres are held fixed. A projectile of mass $m$ is projected from the surface of the sphere of mass $M$ directly towards the centre of the second sphere. The expression for the minimum speed $v$ of the projectile so that it reaches the surface of the second sphere is: