Gravitation · JEE Main previous year questions

The mass density of a planet of radius R varies with the distance r from its centre as $\rho (r)={\rho }_0\left(1-\frac{r^2}{R^2}\right)$. Then the gravitational field is maximum at :

1. A.$r=\frac{1}{\sqrt{3}}R$
2. B.$r=R$
3. C.$r=\sqrt{\frac{3}{4}}R$
4. D.$r=\sqrt{\frac{5}{9}}R$

Planet A has mass M and radius R. Planet B has half the mass and half the radius of Planet A. If the escape velocities from the Planets A and B are ${\mathrm{v}}_{\mathrm{A}}$ and ${\mathrm{v}}_{\mathrm{B}}$, respectively, then $\frac{v_A}{v_B}=\frac{n}{4}$. The value of n is :

1. A.1
2. B.2
3. C.4
4. D.3

1. A.$\frac{2}{3}$
2. B.$\frac{1}{6}$
3. C.$\frac{1}{2}$
4. D.$\frac{1}{3}$

A ball is dropped from the top of a 100 m high tower on a planet. In the last $\frac{1}{2}s$ before hitting the ground, it covers a distance of 19 m. Acceleration due to gravity (in ${\text{ms}}^{-2}$) near the surface on that planet is _______.

An asteroid is moving directly towards the centre of the earth. When at a distance of 10R (R is the radius of the earth) from the earths centre, it has a speed of 12 km/s. Neglecting the effect of earths atmosphere, what will be the speed of the asteroid when it hits the surface of the earth (escape velocity from the earth is 11.2 km/s) ? Give your answer to the nearest integer in kilometer/s _____.

A satellite of mass m is launched vertically upwards with an initial speed u from the surface of the earth. After it reaches height R (R = radius of the earth), it ejects a rocket of mass $\frac{m}{10}$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is (G is the gravitational constant; M is the mass of the earth) :

1. A.$\frac{3m}{8}{\left(u+\sqrt{\frac{5GM}{6R}}\right)}^2$
2. B.$\frac{m}{20}\left(u^2+\frac{113}{100}\frac{GM}{R}\right)$
3. C.$5m\left(u^2-\frac{119}{200}\frac{GM}{R}\right)$
4. D.$\frac{m}{20}{\left(u-\sqrt{\frac{2GM}{3R}}\right)}^2$

The ratio of the weights of a body on the Earth's surface to that on the surface of a planets is $9:4$. The mass of the planet is $\frac{1}{9}$ th of that of the Earth. If 'R' is the radius of the Earth, what is the radius of the planet ? (Take the planets to have the same mass density)

1. A.$\frac{R}{9}$
2. B.$\frac{R}{2}$
3. C.$\frac{R}{3}$
4. D.$\frac{R}{4}$

The value of acceleration due to gravity at Earth's surface is $9.8\mathrm{m}{\mathrm{s}}^{-2}$. The altitude above its surface at which the acceleration due to gravity decreases to $4.9\mathrm{m}{\mathrm{s}}^{-2}$, is close to : (Radius of earth $=6.4\times 10^6\mathrm{m}$)

1. A.$1.6\times 10^6\mathrm{m}$
2. B.$9.0\times 10^6\mathrm{m}$
3. C.$6.4\times 10^6\mathrm{m}$
4. D.$2.6\times 10^6\mathrm{m}$

A spaceship orbits around a planet at a height of 20 km from its surface. Assuming that only gravitational field of the planet acts on the spaceship, what will be the number of complete revolutions made by the spaceship in 24 hours around the planet? [Given ; Mass of planet $=8\times 10^{22}\mathrm{kg}$, Radius of planet $=2\times 10^6\mathrm{m}$, Gravitational constant $\mathrm{G}=6.67\times 10^{-11}\mathrm{N}{\mathrm{m}}^2/k{\mathrm{g}}^2$]

1. A.13
2. B.9
3. C.17
4. D.11

A solid sphere of mass 'M' and radius 'a' is surrounded by a uniform concentric spherical shell of thickness 2a and mass 2M. The gravitational field at distance '3a' from the centre will be :

1. A.$\frac{GM}{3a^2}$
2. B.$\frac{2GM}{9a^2}$
3. C.$\frac{GM}{9a^2}$
4. D.$\frac{2GM}{3a^2}$

A test particle is moving in a circular orbit in the gravitational field produced by a mass density $\rho (r)=\frac{K}{r^2}$. Identify the correct relation between the radius R of the particle's orbit and its period T

1. A.${\mathrm{T}}^2/{\mathrm{R}}^3$ is a constant
2. B.TR is a constant
3. C.$T/{\mathrm{R}}^2$ is a constant
4. D.T/R is a constant

The temperature, at which the root mean square velocity of hydrogen molecules equals their escape velocity from the earth, is closest to : [Boltzmann Constant $k_B=1.38\times 10^{-23}J/K$ Avogadro Number $N_A=6.02\times 10^{26}/kg$ Radius of Earth : $6.4\times 10^6\mathrm{m}$ Gravitational acceleration on Earth = $10\mathrm{m}{\mathrm{s}}^{-2}$]

1. A.$3\times 10^5\mathrm{K}$
2. B.$10^4\mathrm{K}$
3. C.650 K
4. D.800 K

A satellite of mass M is in a circular orbit of radius R about the centre of the earth. A meteorite of the same mass, falling towards the earth, collides with the satellite completely inelastically. The speeds of the satellite and the meteorite are the same, just before the collision. The subsequent motion of the combined body will be :

1. A.in the same circular orbit of radius R
2. B.such that it escapes to infinity
3. C.in a circular orbit of a different radius
4. D.in an elliptical orbit

A straight rod of length L extends from $x=a$ to $x=L+a$. The gravitational force it exerts on a point mass 'm' at $x=0$, if the mass per unit length of the rod is $A+B{x}^2$, is given by :

1. A.$Gm\left[A\left(\frac{1}{a}-\frac{1}{a+L}\right)-BL\right]$
2. B.$Gm\left[A\left(\frac{1}{a}-\frac{1}{a+L}\right)+BL\right]$
3. C.$Gm\left[A\left(\frac{1}{a+L}-\frac{1}{a}\right)+BL\right]$
4. D.$Gm\left[A\left(\frac{1}{a+L}-\frac{1}{a}\right)-BL\right]$

Two satellites, A and B, have masses m and 2m respectively. A is in a circular orbit of radius R, and B is in a circular orbit of radius 2R around the earth. The ratio of their kinetic energies, $T_A/T_B$, is ;

1. A.2
2. B.$\frac{1}{2}$
3. C.$\sqrt{\frac{1}{2}}$
4. D.1

A heavy ball of mass M is suspendeed from the ceiling of a car by a light string of mass m ($m\ll M$). When the car is at rest, the speed of transverse waves in the string is $60{\text{ ms}}^{-1}$. When the car has acceleration a, the wave-speed increases to $60.5{\text{ ms}}^{-1}$. The value of a, in terms of gravitational acceleration g, is closest to :

1. A.$\frac{g}{30}$
2. B.$\frac{g}{5}$
3. C.$\frac{g}{10}$
4. D.$\frac{g}{20}$

The relative uncertainty in the period of a satellite orbiting around the earth is $10^{-2}$. If the relative uncertainty in the radius of the orbit is negligible, the relative uncertainty in the mass of the earth is :

1. A.$10^{-2}$
2. B.$2\times 10^{-2}$
3. C.$3\times 10^{-2}$
4. D.$6\times 10^{-2}$

A body of mass m is moving in a circular orbit of radius R about a planet of mass M. At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius $\frac{R}{2}$, and the other mass, in a circular orbit of radius $\frac{3R}{2}$. The difference between the final and initial total energies is :

1. A.$-\frac{GMm}{2R}$
2. B.$+\frac{GMm}{6R}$
3. C.$\frac{GMm}{2R}$
4. D.$-\frac{GMm}{6R}$

A satellite is revolving in a circular orbit at a height '$h$' from the earth's surface (radius of earth $R$; $h\ll R$). The minimum increase in its orbital velocity required, so that the satellite could escape from the earth's gravitational field, is close to : (Neglect the effect of atmosphere.)

1. A.$\sqrt{2gR}$
2. B.$\sqrt{gR}$
3. C.$\sqrt{gR/2}$
4. D.$\sqrt{gR}(\sqrt{2}-1)$

1. A.$\frac{-2GM}{3R}$
2. B.$\frac{-2GM}{R}$
3. C.$\frac{-GM}{2R}$
4. D.$\frac{-GM}{R}$

1. A.$2\sqrt{\frac{\pi gL}{{\mu }_0}\tan \theta }$
2. B.$\sqrt{\frac{\pi \lambda gL}{{\mu }_0}\tan \theta }$
3. C.$\sin \theta \sqrt{\frac{\pi \lambda gL}{{\mu }_0\cos \theta }}$
4. D.$2\sin \theta \sqrt{\frac{\pi \lambda gL}{{\mu }_0\cos \theta }}$

Four particles, each of mass $M$ and equidistant from each other, move along a circle of radius $R$ under the action of their mutual gravitational attraction. The speed of each particle is :

1. A.$\sqrt{\frac{GM}{R}}$
2. B.$\sqrt{2\sqrt{2}\frac{GM}{R}}$
3. C.$\sqrt{\frac{GM}{R}\left(1+2\sqrt{2}\right)}$
4. D.$\frac{1}{2}\sqrt{\frac{GM}{R}\left(1+2\sqrt{2}\right)}$

A satellite moving with velocity $v$ in a force free space collects stationary interplanetary dust at a rate of $\frac{dM}{dt}=\alpha v$ where $M$ is the mass (of satellite + dust) at that instant. The instantaneous acceleration of the satellite is

1. A.$-\frac{\alpha v^2}{2M}$
2. B.$-\frac{\alpha v^2}{M}$
3. C.$-\alpha v^2$
4. D.$-\frac{2\alpha v^2}{M}$

A point particle is held on the axis of a ring of mass $m$ and radius $r$ at a distance $r$ from its centre $C$. When released, it reaches $C$ under the gravitational attraction of the ring. Its speed at $C$ will be

1. A.$\sqrt{\frac{2Gm}{r}(\sqrt{2}-1)}$
2. B.$\sqrt{\frac{Gm}{r}}$
3. C.$\sqrt{\frac{2Gm}{r}\left(1-\frac{1}{\sqrt{2}}\right)}$
4. D.$\sqrt{\frac{2Gm}{r}}$

The change in the value of $g$ at a height $h$ above the surface of the earth is the same as at a depth $d$ below the surface of earth. When both $d$ and $h$ are much smaller than the radius of earth, then which one of the following is correct?

1. A.$d=\frac{3h}{2}$
2. B.$d=\frac{h}{2}$
3. C.$d=h$
4. D.$d=2h$