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. \(2Gm^{2} \hat{j}\)$\stackrel{}{}$

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: