Question

Two uniform solid spheres of masses $M$ and $4M$ but of equal radii $R$ , have a centre to centre separation $6R$ , as depicted in the figure. A projectile of mass $m$ is projected from the surface of the sphere of mass $M$ directly towards the center of the second sphere. What should be the minimum speed of projection so that it reaches the surface of the second sphere?

• A. $\sqrt{\frac{4}{5} \frac{G M}{R}}$
• B. $\sqrt{\frac{5}{4} \frac{G M}{R}}$
• C. $\sqrt{\frac{3}{5} \frac{G M}{R}}$
• D. $\sqrt{\frac{5}{3} \frac{G M}{R}}$

Answer 1

Answer: (C)

Let the projectile of mass $m$ be fired with minimum velocity, $v$ from the surface of sphere of mass $M$ to reach the surface of sphere of mass $4M$ . Let $N$ be neutral point at a distance $r$ from the centre of the sphere of mass $M$ .
At neutral point $N$ ,
$\frac{G M m}{r^{2}}=\frac{G \left(\right. 4 M \left.\right) m}{\left(\right. 6 R - r \left.\right)^{2}}$
$\left(\right. 6 R - r \left.\right)^{2}=4r^{2}$
$6R-r=\pm2r$ or $r=2R \, $ or $-6R$
The point $r=-6R$ does not concern us.
Thus, $ON \, = \, r \, = \, 2R.$
It is sufficient to project the projectile with a speed which would enable it to reach $N$ . Thereafter, the greater gravitational pull of $4M$ would suffice.
The mechanical energy at the surface of $M$ is
$E_{i}=\frac{1}{2}mv^{2}-\frac{G M m}{R}-\frac{G \left(\right. 4 M \left.\right) m}{5 R}$
At the neutral point $N$ , the speed approaches zero.
The mechanical energy at $N$ is
$E_{N}=-\frac{G M m}{2 R}-\frac{G \left(\right. 4 M \left.\right) m}{4 R}=-\frac{G M m}{2 R}-\frac{G M m}{R}$
According to law of conservation of mechanical energy,
$E_{i}=E_{N}$
$\frac{1}{2}mv^{2}-\frac{G M m}{R}-\frac{4 G M m}{5 R}=-\frac{G M m}{2 R}-\frac{G M m}{R}$