Gravitation · JEE Main previous year questions

1. A.0.33
2. B.0.67
3. C.1
4. D.0.5

Match List - I with List - II. $$\begin{array}{llll}\text{List - I}& & \text{List - II}& \\ \text{A.}& \text{Boltzmann constant}& \text{I.}& [{\mathrm{M}}^{-1}{\mathrm{L}}^3{\mathrm{T}}^{-2}]\\ \text{B.}& \text{Stefan’s constant}& \text{II.}& [\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-1}]\\ \text{C.}& \text{Planck’s constant}& \text{III.}& [\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-2}{\mathrm{K}}^{-1}]\\ \text{D.}& \text{Gravitational constant}& \text{IV.}& [\mathrm{M}{\mathrm{L}}^0{\mathrm{T}}^{-3}{\mathrm{K}}^{-4}]\end{array}$$ Choose the correct answer from the options given below :

1. A.A-I, B-II, C-III, D-IV
2. B.A-IV, B-III, C-II, D-I
3. C.A-III, B-IV, C-II, D-I
4. D.A-II, B-I, C-IV, D-III

1. A.A-II, B-IV, C-I, D-III
2. B.A-IV, B-II, C-I, D-III
3. C.A-IV, B-I, C-II, D-III
4. D.A-III, B-I, C-II, D-IV

A planet $\left({\mathrm{P}}_1\right)$ is moving around the star of mass $2\mathrm{M}$ in the orbit of radius $R$. Another planet $\left({\mathrm{P}}_2\right)$ is moving around another star of mass $4\mathrm{M}$ in a orbit of radius $2R$. Ratio of time periods of revolution of ${\mathrm{P}}_2$ and ${\mathrm{P}}_1$ is ____.

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

Three masses $200\mathrm{kg}$, $300\mathrm{kg}$ and $400\mathrm{kg}$ are placed at the vertices of an equilateral triangle with sides $20\mathrm{m}$ . They are rearranged on the vertices of a bigger triangle of side $25\mathrm{m}$ and with the same centre. The work done in this process ____ J. (Gravitational constant $\mathrm{G}=6.7\times 10^{-11}\mathrm{N}{\mathrm{m}}^2/k{\mathrm{g}}^2$ )

1. A.$4.77\times 10^{-7}$
2. B.$1.74\times 10^{-7}$
3. C.$9.86\times 10^{-6}$
4. D.$2.85\times 10^{-7}$

A block is sliding down with constant velocity on an inclined plane of slope $\theta$ and at an instant $t=0$ this block is given an upward momentum so that it starts moving up on the inclined surface with velocity $u$. The distance $(S)$ travelled by the block before its velocity become zero, is _____. ($g=$ gravitational acceleration)

1. A.$\frac{2u^2}{g\cos \theta }$
2. B.$\frac{u^2}{\sqrt{2}g\cos \theta }$
3. C.$\frac{u^2}{4g\sin \theta }$
4. D.$\frac{u^2}{2g\cos \theta }$

The escape velocity from a spherical planet $A$ is $10km/s$. The escape velocity from another planet $B$ whose density and radius are $10\%$ of those of planet $A$, is ____ $m/s$.

1. A.1000
2. B.$200\sqrt{5}$
3. C.$1000\sqrt{2}$
4. D.$100\sqrt{10}$

1. A.2
2. B.$2\sqrt{5}$
3. C.1
4. D.3

Initially a satellite of 100 kg is in a circular orbit of radius $1.5{\mathrm{R}}_{\mathrm{E}}$. This satellite can be moved to a circular orbit of radius $3R_E$ by supplying $\alpha \times 10^6\mathrm{J}$ of energy. The value of $\alpha$ is ____. (Take Radius of Earth $R_E=6\times 10^6\mathrm{m}$ and $\mathrm{g}=10m/{\mathrm{s}}^2$)

1. A.500
2. B.1000
3. C.100
4. D.150

A spherical body of radius $r$ and density $\sigma$ falls freely through a viscous liquid having density $\rho$ and viscosity $\eta$ and attains a terminal velocity $v_0$. Estimated maximum error in the quantity $\eta$ is : (Ignore errors associated with $\sigma ,\rho$ and $g$, gravitational acceleration)

1. A.$2\left[\frac{\Delta r}{r}-\frac{\Delta v_0}{v_0}\right]$
2. B.$2\left[\frac{\Delta r}{r}+\frac{\Delta v_0}{v_0}\right]$
3. C.$\frac{2\Delta r}{r}+\frac{\Delta v_0}{v_0}$
4. D.$2\frac{\Delta r}{r}-\frac{\Delta v_0}{v_0}$

Consider two identical metallic spheres of radius $R$ each having charge $Q$ and mass $m$. Their centers have an initial separation of $4R$. Both the spheres are given an initial speed of $u$ towards each other. The minimum value of $u$, so that they can just touch each other is: (Take $k=\frac{1}{4\pi {ϵ}_0}$ and assume $k{Q}^2>G{m}^2$ where $G$ is the Gravitational constant)

1. A.$\sqrt{\frac{k{Q}^2}{4mR}\left(1+\frac{G{m}^2}{k{Q}^2}\right)}$
2. B.$\sqrt{\frac{k{Q}^2}{2mR}\left(1-\frac{G{m}^2}{2k{Q}^2}\right)}$
3. C.$\sqrt{\frac{k{Q}^2}{2mR}\left(1-\frac{G{m}^2}{k{Q}^2}\right)}$
4. D.$\sqrt{\frac{k{Q}^2}{4mR}\left(1-\frac{G{m}^2}{k{Q}^2}\right)}$

An object is kept at rest at a distance of $3R$ above the earth's surface where $R$ is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is : (Assume $\mathrm{M}=$ mass of earth, $\mathrm{G}=$ Universal gravitational constant)

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

Match the LIST-I with LIST-II $$\begin{array}{llll}\text{List - I}& & \text{List - II}& \\ \text{A.}& \text{Gravitational constant}& \text{I.}& \left[\mathrm{L}{\mathrm{T}}^{-2}\right]\\ \text{B.}& \text{Gravitational potential energy}& \text{II.}& \left[{\mathrm{L}}^2{\mathrm{T}}^{-2}\right]\\ \text{C.}& \text{Gravitational potential}& \text{III.}& \left[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-2}\right]\\ \text{D.}& \text{Acceleration due to gravity}& \text{IV.}& \left[{\mathrm{M}}^{-1}{\mathrm{L}}^3{\mathrm{T}}^{-2}\right]\end{array}$$ Choose the correct answer from the options given below:

1. A.A-IV, B-III, C-II, D-I
2. B.A-II, B-IV, C-III, D-I
3. C.A-I, B-III, C-IV, D-II
4. D.A-III, B-II, C-I, D-IV

Three identical spheres of mass m , are placed at the vertices of an equilateral triangle of length $a$. When released, they interact only through gravitational force and collide after a time $\mathrm{T}=4\mathrm{seconds}$. If the sides of the triangle are increased to length $2a$ and also the masses of the spheres are made 2 m , then they will collide after____seconds.

A satellite of mass $1000\mathrm{kg}$ is launched to revolve around the earth in an orbit at a height of $270\mathrm{km}$ from the earth's surface. Kinetic energy of the satellite in this orbit is ____ $\times 10^{10}\mathrm{J}$. (Mass of earth $=6\times 10^{24}\mathrm{kg}$, Radius of earth $=6.4\times 10^6\mathrm{m}$, Gravitational constant $=6.67\times 10^{-11}\mathrm{N}{\mathrm{m}}^2\mathrm{k}{\mathrm{g}}^{-2}$ )

A satellite is launched into a circular orbit of radius ' R ' around the earth. A second satellite is launched into an orbit of radius $1.03\mathrm{R}$ . The time period of revolution of the second satellite is larger than the first one approximately by

1. A.$3\%$
2. B.$2.5\%$
3. C.$4.5\%$
4. D.$9\%$

If a satellite orbiting the Earth is 9 times closer to the Earth than the Moon, what is the time period of rotation of the satellite? Given rotational time period of Moon $=27$ days and gravitational attraction between the satellite and the moon is neglected.

1. A.3 days
2. B.27 days
3. C.81 days
4. D.1 day

A satellite of mass $\frac{M}{2}$ is revolving around earth in a circular orbit at a height of $\frac{R}{3}$ from earth surface. The angular momentum of the satellite is $\mathrm{M}\sqrt{\frac{\mathrm{GMR}}{x}}$. The value of $x$ is _________ , where M and R are the mass and radius of earth, respectively. ( G is the gravitational constant)

1. A.$11:10$
2. B.$12:11$
3. C.$16:9$
4. D.$12:9$

A satellite of $10^3\mathrm{kg}$ mass is revolving in circular orbit of radius $2R$. If $\frac{10^{4}R}{6}\mathrm{J}$ energy is supplied to the satellite, it would revolve in a new circular orbit of radius (use $g=10m/{\mathrm{s}}^2$, $R=$ radius of earth)

1. A.$4\mathrm{R}$
2. B.$6\mathrm{R}$
3. C.$2.5\mathrm{R}$
4. D.$3\mathrm{R}$

Two satellite A and B go round a planet in circular orbits having radii 4R and R respectively. If the speed of A is $3v$, the speed of B will be :

1. A.$6v$
2. B.$\frac{4}{3}v$
3. C.$3v$
4. D.$12v$

A satellite revolving around a planet in stationary orbit has time period 6 hours. The mass of planet is one-fourth the mass of earth. The radius of orbit of the satellite is : (Given $=$ Radius of geo-stationary orbit for earth is $4.2\times 10^4\mathrm{km}$)

1. A.$1.68\times 10^5\mathrm{km}$
2. B.$1.4\times 10^4\mathrm{km}$
3. C.$8.4\times 10^4\mathrm{km}$
4. D.$1.05\times 10^4\mathrm{km}$

A metal wire of uniform mass density having length $L$ and mass $M$ is bent to form a semicircular arc and a particle of mass $\mathrm{m}$ is placed at the centre of the arc. The gravitational force on the particle by the wire is :

1. A.$\frac{\mathrm{GmM}{\pi }^2}{{\mathrm{L}}^2}$
2. B.$\frac{\mathrm{GmM}\pi }{2{\mathrm{L}}^2}$
3. C.0
4. D.$\frac{2\mathrm{GmM}\pi }{{\mathrm{L}}^2}$

A $90\text{ kg}$ body placed at $2\text{R}$ distance from surface of earth experiences gravitational pull of : $(\text{R}=\text{Radius of earth},g=10{\text{ m s}}^{-2})$

1. A.$300\text{ N}$
2. B.$225\text{ N}$
3. C.$100\text{ N}$
4. D.$120\text{ N}$

Correct formula for height of a satellite from earth's surface is :

1. A.${\left(\frac{T^2{R}^{2}g}{4{\pi }^2}\right)}^{1/3}-R$
2. B.${\left(\frac{T^2{R}^{2}g}{4\pi }\right)}^{1/2}-R$
3. C.${\left(\frac{T^2{R}^{2}g}{4{\pi }^2}\right)}^{-1/3}+R$
4. D.${\left(\frac{T^2{R}^2}{4{\pi }^{2}g}\right)}^{1/3}-R$