Gravitational Potential Energy of a Two Mass System

Q.

The potential energy curve for the$H_2$ molecule as a function of internuclear distance is:

1. 

2. 

3. 

4. 

Q. Two uniform solid spheres held fixed of equal radii $R$, but mass $M$ and $4M$ have centre to centre separation $6R$. A projectile of mass $m$ is projected from the surface of the sphere of mass $M$ directly towards the centre of the second sphere. Obtain an expression for the minimum speed $v$ of the projectile so that it reaches the surface of the second sphere.

1. $(\frac{3GM}{5R}{)}^{1/2}$
2. $(\frac{5GM}{3R}{)}^{1/2}$
3. $(\frac{2GM}{5R}{)}^{1/2}$
4. $(\frac{5GM}{2R}{)}^{1/2}$

Q.

The minimum and maximum distances of a satellite from the center of the Earth are 2R and 4R respectively, where R is the radius of Earth and M is the mass of Earth. The radius of curvature at the point of minimum distance is

Q. Two bodies of masses $m_1$ and $m_2$ are initially at rest at an infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is

1. ${\left[2G\frac{(m_1-m_2)}{r}\right]}^{1/2}$
2. ${\left[\frac{2G}{r}\right(m_1+m_2\left)\right]}^{1/2}$
3. ${\left[\frac{r}{2G\left(m_1{m}_2\right)}\right]}^{1/2}$
4. ${\left[\frac{2G}{r}{m}_1{m}_2\right]}^{1/2}$

Q. Two bodies of masses $m_1$ and $m_2$ are initially at rest at an infinite distance apart. They are then allowed to move towards each other under mutual gravitational attraction. Their relative velocity of approach at a separation distance $r$ between them is

1. ${\left[2G\frac{(m_1-m_2)}{r}\right]}^{1/2}$
2. ${\left[\frac{2G}{r}\right(m_1+m_2\left)\right]}^{1/2}$
3. ${\left[\frac{r}{2G\left(m_1{m}_2\right)}\right]}^{1/2}$
4. ${\left[\frac{2G}{r}{m}_1{m}_2\right]}^{1/2}$

Q. The minimum energy required to launch a $m\text{kg}$ satellite from the earth’s surface in a circular orbit at an altitude $2R$, where $R$ is the radius of earth is

1. $\frac{5}{3}mgR$
2. $\frac{4}{3}mgR$
3. $\frac{5}{6}mgR$
4. $\frac{5}{4}mgR$

Q. Inside a fixed sphere of radius $R$ and uniform density $\rho$, there is spherical cavity of radius $\frac{R}{2}$ such that surface of the cavity passes through the centre of the sphere as shown in figure. A particle of mass $m$ is released from rest at centre $\text{B}$ of the cavity. Calculate velocity with which particle strikes the centre of the sphere. Neglect earth's gravity. Initially sphere and particle are at rest.

1. $\sqrt{\frac{2}{3}{\pi }^2{G}^2{\rho }^2{R}^2}$
2. $\sqrt{\frac{2}{3}\pi G{\rho }^2{R}^2}$
3. $\sqrt{\frac{2}{3}\pi G\rho R^2}$
4. $\sqrt{\frac{2}{5}\pi G\rho R^2}$

Q. A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth's gravitational influence? Mass of the satellite $=200kg$; mass of the earth $=6.0\times {10}^{24}kg$; radius of the earth $=6.4\times {10}^{6}m$; G = $6.67\times {10}^{-11}N{m}^{2}k{g}^{-2}$.