Gravitational Potential Energy of a Two Mass System

Q. In space two point masses of mass $10kg$ each are fixed. The masses are separated by a distance $10m$. Another point mass of mass $1kg$ is to be projected from the point at midpoint of line joining the fixed masses such that it escapes to infinity. The minimum speed of the projection is

1. $\sqrt{G}\frac{m}{s}$
2. $\sqrt{2G}\frac{m}{s}$
3. $\sqrt{8G}\frac{m}{s}$
4. $\sqrt{6G}\frac{m}{s}$

Q. The energy required to take a satellite to a height $h$ above Earth surface (radius of Earth $=6.4\times {10}^3\text{km}$) is $E_1$and kinetic energy required for the satellite to be in a circular orbit at this height is $E_2$. The value of $h$ for which $E_1$ and $E_2$ are equal is

1. $6.4\times {10}^3\text{km}$
2. $3.2\times {10}^3\text{km}$
3. $1.6\times {10}^3\text{km}$
4. $28\times {10}^4\text{km}$

Q. Two particles of masses $m$ & $M$ are initially at rest at an infinite distance apart. They move towards each other and gain speeds due to gravitational attraction. Find their speeds when separation between the masses becomes equal to $d$.

1. $\sqrt{\frac{G{m}^2}{d(m+M)}},\sqrt{\frac{G{m}^2}{d(m+M)}}$
2. $\sqrt{\frac{2G{M}^2}{d(m+M)}},\sqrt{\frac{2G{m}^2}{d(m+M)}}$
3. $\sqrt{\frac{1}{2}\frac{G{M}^2}{d(m+M)}},\sqrt{\frac{1}{2}\frac{G{m}^2}{d(m+M)}}$
4. $2\sqrt{\frac{G{M}^2}{d(m+M)}},2\sqrt{\frac{G{m}^2}{d(m+M)}}$
   

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. A particle of mass $m$ is kept at rest at a height $3R$ from the surface of earth, where $R$ is radius of earth and $M$ is mass of earth. The minimum speed with which it should be projected, so that it does not return back, is ($g$ is acceleration due to gravity on the surface of earth)

1. $\sqrt{\left(\frac{GM}{R}\right)}$
2. $\sqrt{\left(\frac{GM}{2R}\right)}$
   
3. $\sqrt{\left(\frac{2GM}{R}\right)}$
4. $\sqrt{\left(\frac{GM}{3R}\right)}$

Q. Energy of a satellite in circular orbit is $-E_0$. The energy required to move the satellite to a circular orbit of $3$ times the radius of the initial orbit is

1. $\frac{2}{3}{E}_0$
2. $2E_0$
3. $\frac{E_0}{3}$
4. $\frac{3}{2}{E}_0$

Q. An artificial satellite moving in a circular orbit around the earth has a total (kinetic + potential) energy $E_0$. Its potential energy is

1. 
2. 
3. 
4. 

Q. A particle of mass $m$ is thrown upwards from the surface of the earth, with a velocity $u$. The mass and the radius of the earth are, respectively, $M$ and $R$. $G$ is gravitational constant and $g$ is acceleration due to gravity on the surface of the earth. The minimum value of $u$ so that the particle does not return back to the earth is

1. $\sqrt{2g{R}^2}$
2. $\sqrt{\frac{2GM}{R^2}}$
3. $\sqrt{\frac{2GM}{R}}$
4. $\sqrt{\frac{2gM}{R^2}}$

Q. The additional kinetic energy to be provided to a satellite of mass $m$ revolving around a planet of mass $M$ to transfer it from a circular orbit of radius $R_1$ to another circular orbit of radius $R_2(R_2>R_1)$ is

1. $GMm(\frac{1}{R_1}+\frac{1}{R_2})$
2. $GMm(\frac{1}{R_1}-\frac{1}{R_2})$
3. $2GMm(\frac{1}{R_1}-\frac{1}{R_2})$
4. $\frac{1}{2}GMm(\frac{1}{R_1}-\frac{1}{R_2})$

Q. A satellite is orbiting around the earth in a circular orbit. Its orbital speed is $V_0$. A rocket on board is fired from the satellite which imparts a thrust to the satellite directed radially away from the centre of the earth. The duration of the engine burn is negligible so that it can be considered instantaneous. Due to this thrust, a velocity variation $\Delta V$ is imparted to the satellite. The minimum value of the ratio $\frac{\Delta V}{V_0}$ for which the satellite will escape out of the gravitational field of the earth is

Q.

A particle is fired vertically upward with a speed of 15 km $s^{-1}$ With what speed will it move in intersteller space. Assume only earth's gravitational field.

Q. The given plot shows the variation of $U$, the potential energy of interaction between two particles w.r.t. the distance of separation, $r$.


$1.$ <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $\text{B}$ and $\text{D}$ are equilibrium points.
$2.$ <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $\text{C}$ is a point of stable equilibrium.
$3.$ The force of interaction between the two particles is attractive between points <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $\text{C}$ and $\text{D}$ and repulsive between <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $\text{D}$ and $\text{E}$.
$4.$ The force of interaction between particles is repulsive between points <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $\text{E}$ and $\text{F}$.

Which of the above statements are correct?

1. $1$ and $2$
2. $1$ and $4$
3. $2$ and $4$
4. $2$ and $3$

Q. Two masses $M_1$ & $M_2$ are initially at rest and are separated by a very large distance. If the masses approach each other subsequently due to gravitational attraction between them, their relative velocity of approach at a separation $d$ is

1. $\sqrt{\frac{2G(M_1+M_2)}{d}}$
2. $\sqrt{\frac{4G(M_1+M_2)}{d}}$
3. $\sqrt{\frac{4G\left(M_1{M}_2\right)}{d}}$
4. $\sqrt{\frac{G(M_1+M_2)}{d}}$

Q. Two particles $\text{A}$ and $\text{B}$ have masses $m$ and $2m$ respectively. They are held at separation $r_0$ in space. $\text{A}$ is given a velocity $v_0$ along the line joining the two masses (away from $\text{B}$) and $\text{B}$ is released simultaneously. Find the range of velocity $v_0$ for which the two particles would remain bound under their mutual gravitation.

1. $v_0>\sqrt{\frac{6Gm}{r_0}}$
   
2. $v_0<\sqrt{\frac{6Gm}{r_0}}$
   
3. $v_0<\sqrt{\frac{2Gm}{r_0}}$
   
4. $v_0<\sqrt{\frac{12Gm}{r_0}}$
   

Q. What is the minimum energy required to launch a satellite of mass $m$ from the surface of a planet of mass $M$ and radius $R$ in a circular orbit at an altitude of $2R$ ?

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

Q.

There are two point masses each of mass m are at the ends of a rod of mass m. The rod is released. What will be the angular speed when it becomes vertical?

1. 

2. 

3. 

4. 

Q. How many astronomical units (AU) are there in 2 meter ?

1. $6.67\times {10}^{-12}AU$
2. $13.34\times {10}^{-11}AU$
3. $6.67\times {10}^{-11}AU$
4. $13.34\times {10}^{-12}AU$

Q. If the potential energy between two molecules is given by $U=-\frac{A}{r^6}+\frac{B}{r^{12}},$ then at equilibrium, separation between molecules and the potential energy respectively are:

1. ${\left(\frac{B}{2A}\right)}^{\frac{1}{6}},-\left(\frac{A^2}{2B}\right)$
2. ${\left(\frac{B}{A}\right)}^{\frac{1}{6}},0$
3. ${\left(\frac{2B}{A}\right)}^{\frac{1}{6}},-\left(\frac{A^2}{4B}\right)$
4. ${\left(\frac{2B}{A}\right)}^{\frac{1}{6}},\left(\frac{A^2}{2B}\right)$

Q. A particle is fired vertically upwards with a speed of $9.8km/s$. Find the maximum height attained by the particle. Radius of earth = $6400km$ and $g$ at the surface = $9.8{m/s}^2$. Consider only earth’s gravitation.

1. $50100km$
2. $20900km$
3. $43200km$
4. $35200km$

Q. Two bodies of masses $m_1$ and $m_2$ are initially at rest at 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. 
2. 
3. 
4. 

Q. A space station is set up in space at a distance equal to the earth’s radius from the surface of the earth. Suppose a satellite can be launched from the space station. Let $v_1$ and $v_2$ be the escape velocities if the satellite on the earth’s surface and space station, respectively. Then,

1. $v_2=v_1$
2. $v_2<v_1$
3. $v_2>v_1$
4. (a), (b) and (c) are valid depending on satellite.

Q. A particle is fired vertically upwards with a speed of 15 $km/s$. With what speed (in $km/s$) will it move in interstellar space? Assume only earth’s gravitational field.

Q. Two masses $M_1$ & $M_2$ are initially at rest and are separated by a very large distance. If the masses approach each other subsequently due to gravitational attraction between them, their relative velocity of approach at a separation $d$ is

1. $\sqrt{\frac{2G(M_1+M_2)}{d}}$
2. $\sqrt{\frac{4G(M_1+M_2)}{d}}$
3. $\sqrt{\frac{4G\left(M_1{M}_2\right)}{d}}$
4. $\sqrt{\frac{G(M_1+M_2)}{d}}$

Q. The physical quantities not having same dimensions are :

1. torque and work
2. momentum and Plancks constant
3. stress and Youngs modulus
4. speed and ${\left({\mu }_0{ϵ}_0\right)}^{-1/2}$

Q. Two particles of mass $m_1$ and $m_2$ are initially at rest at infinite distance. Find their velocity of approach due to gravitational attraction, when separation is d :

1. $\sqrt{\frac{2G(m_1+m_2)}{d}}$
2. $\sqrt{\frac{G(2m_1+m_2)}{3d}}$
3. $\sqrt{\frac{3G(2m_1+m_2)}{d}}$
4. $\sqrt{\frac{G(m_1+m_2)}{d}}$

Q. A satellite of mass $1000\text{kg}$ is rotating around the earth in a circular orbit of radius $3R$. What extra energy should be given to this satellite if it is to be lifted into an orbit of radius $4R$ ?

1. $2.614\times {10}^9\text{J}$
2. $6\times {10}^8\text{J}$
3. $3\times {10}^8\text{J}$
4. $5.3\times {10}^9\text{J}$

Q. A system consists of 8 partices of different masses. How many terms appear in expression of potential energy of system?

1. 8
2. 16
3. 28
4. 56

Q. The orbital velocity of a body at higher $h$ above the surface of the Earth is $36$% of that near the surface of the Earth of radius $R$. If the escape velocity at the surface of Earth is $11.2km{s}^{-1}$, then its value at the height $h$ will be:-

1. $11.2km{s}^{-1}$
2. $\sqrt{\frac{h}{R}}\times 11.2km{s}^{-1}$
3. $\frac{9}{25}\times 11.2km{s}^{-1}$
4. $\sqrt{\frac{R}{h}}\times 11.2km{s}^{-1}$

Q. A satellite orbiting the earth in a circular orbit of radius $R$ completes one revolution in $3h$. If orbital radius of geostationary satellite is $36000km$, orbital radius of satellite is

1. $9000km$
2. $15000km$
3. $12000km$
4. $6000km$

Q. Two masses $M_1$ & $M_2$ are initially at rest and are separated by a very large distance. If the masses approach each other subsequently due to gravitational attraction between them, their relative velocity of approach at a separation $d$ is

1. $\sqrt{\frac{2G(M_1+M_2)}{d}}$
2. $\sqrt{\frac{4G(M_1+M_2)}{d}}$
3. $\sqrt{\frac{4G\left(M_1{M}_2\right)}{d}}$
4. $\sqrt{\frac{G(M_1+M_2)}{d}}$