Chapter 7

The Fig. \(7.9\) shows a planet in elliptical orbit around the sun \(S\). Where is the kinetic energy of the planet maximum? (A) \(P_{1}\) (B) \(P_{2}\) (C) \(P_{3}\) (D) \(P_{4}\)

The ratio of the radii of the planets \(P_{1}\) and \(P_{2}\) is \(k_{1}\). The ratio of the acceleration due to the gravity on them is \(k_{2}\). The ratio of the escape velocities from them will be (A) \(k_{1} k_{2}\) (B) \(\sqrt{k_{1} k_{2}}\) (C) \(\sqrt{\left(k_{1} / k_{2}\right)}\) (D) \(\sqrt{\left(k_{2} / k_{1}\right)}\)

The orbital speed of Jupiter is (A) greater than the orbital speed of earth. (B) lesser than the orbital speed of earth. (C) equal to the orbital speed of earth. (D) Zero.

The period of a satellite in a circular orbit of radius \(R\) is \(T\). The period of another satellite in a circular orbit of radius \(4 R\) is (A) \(4 T\) (B) \(T / 4\) (C) \(8 \mathrm{~T}\) (D) \(T / 8\)

The period of a satellite in a circular orbit around a planet is independent of, (A) the mass of the planet. (B) the radius of the planet. (C) the mass of the satellite. (D) all of three parameters \(a, b\) and \(c\).

The acceleration due to gravity on the surface of the moon is \(\frac{1}{6}\) that of the surface of earth and the diameter of the moon is \(\frac{1}{4}\) that of earth. The ratio of escape velocities on earth and moon will be (A) \(\frac{\sqrt{6}}{2}\) (B) \(\sqrt{24}\) (C) 3 (D) \(\frac{\sqrt{3}}{2}\)

At a height above the surface of the earth equal to the radius of the earth the value of \(g\) (acceleration due to gravity on the surface of the earth) will be nearly (A) Zero (B) \(\sqrt{g}\) (C) \(\frac{g}{4}\) (D) \(\frac{g}{2}\)

Two satellites \(S_{1}\) and \(S_{2}\) describe circular orbits of radii \(r\) and \(2 r\) respectively around a planet. If the orbital angular velocity of \(S_{1}\) is \(\omega\), the orbital angular velocity of \(S_{2}\) is (A) \(\frac{\omega}{2 \sqrt{2}}\) (B) \(\frac{\omega \sqrt{2}}{3}\) (C) \(\frac{\omega}{\sqrt{2}}\) (D) \(\omega \sqrt{2}\)

A person brings a mass of \(1 \mathrm{~kg}\) from infinity to a point \(A\). Initially the mass was at rest but it moves at a speed of \(2 \mathrm{~m} / \mathrm{s}\) as it reaches \(A\). The work done by the person on the mass is \(-3 \mathrm{~J}\). The potential at \(A\) is (A) \(-3 \mathrm{~J} / \mathrm{kg}\) (B) \(-2 \mathrm{~J} / \mathrm{kg}\) (C) \(-5 \mathrm{~J} / \mathrm{kg}\) (D) None of these

An artificial satellite moving in a circular orbit around the earth has a total energy \((\mathrm{KE}+\mathrm{PE})\) is \(E_{0}\). Its potential energy is (A) \(-E_{0}\) (B) \(1.5 E_{0}\) (C) \(2 E_{0}\) (D) \(E_{0}\)

Four particles of equal mass \(M\) move along a circle of radius \(R\) under the action of their mutual gravitational attraction. The speed of each particle is (A) \(\frac{G M}{R}\) (B) \(\sqrt{\left(\frac{G M}{R}\right)}\) (C) \(\sqrt{\left[\frac{G M}{R}\left(\frac{2 \sqrt{2}+1}{4}\right)\right]}\) (D) \(\sqrt{\left[\frac{G M}{R}(\sqrt{2}+1)\right]}\)

A simple pendulum has a time period \(T_{1}\) when on the earth's surface, and \(T_{2}\) when taken to a height \(R\) above the earth's surface, where \(R\) is radius of earth. The value of \(T_{2} / T_{1}\) is (A) 1 (B) \(\sqrt{2}\) (C) 4 (D) 2

A uniform spherical shell gradually shrinks maintaining its shape. The gravitational potential at the centre (A) Increases (B) Decreases (C) Remains constant (D) Oscillates

A body of mass \(m\) rises to a height \(h=R / 5\) from the earth's surface where \(R\) is radius of the earth. If \(g\) is acceleration due to gravity at the earth surface, the increase in potential energy is (A) \(m g R\) (B) \((4 / 5) m g R\) (C) \((1 / 6) m g R\) (D) \((6 / 7) m g R\)

Imagine a light planet revolving around a very massive star in a circular orbit of radius \(R\) with a period of revolution \(T\). If the gravitational force of attraction between the planet and the star is proportional to \(R^{-5 / 2}\), \(T^{2}\) is proportional to (A) \(R^{3}\) (B) \(R^{7 / 2}\) (C) \(R^{3 / 2}\) (D) \(R^{3.75}\)

A body is suspended from a spring balance kept in a satellite. The reading of the balance is \(W_{1}\) when the satellite goes in an orbit of radius \(R\) and is \(W_{2}\) when it goes in an orbit of radius \(2 R\). (A) \(W_{1}=W_{2}\) (B) \(W_{1}W_{2}\) (D) \(W_{1} \neq W_{2}\)

A planet is revolving around the sun in elliptical orbit. Its closest distance from the sun is \(r\) and the farthest distance is \(R\). If the orbital velocity of the planet closest to the sun be \(v\), then what is the velocity at the farthest point? (A) \(\overline{v r} / R\) (B) \(v R / r\) (C) \(v\left(\frac{r}{R}\right)^{1 / 2}\) (D) \(v\left(\frac{R}{r}\right)^{1 / 2}\)

The orbital velocity of an artificial satellite in a circular orbit just above earth's surface is \(v_{0} .\) For a satellite orbiting in a circular orbit at an altitude of half of earth's radius is (A) \(\sqrt{\frac{3}{2}} v_{0}\) (B) \(\sqrt{\frac{2}{3}} v_{0}\) (C) \(\frac{3}{2} v_{0}\) (D) \(\frac{2}{3} v_{0}\)

A particle is placed in a field characterized by a value of gravitational potential given by \(V=-k x y\), where \(k\) is a constant. If \(\vec{E}_{g}\) is the gravitational field then, (A) \(\vec{E}_{g}=k(x \hat{i}+y \hat{j})\) and is conservative in nature. (B) \(\vec{E}_{g}=k(y \hat{i}+x \hat{j})\) and is conservative in nature. (C) \(\vec{E}_{g}=k(x \hat{i}+y \hat{j})\) and is non-conservative in nature (D) \(\vec{E}_{g}=k(y \hat{i}+x \hat{j})\) and is non-conservative in nature.

Three equal masses \(m \mathrm{~kg}\) are placed at the vertices of an equilateral triangle of side \(a\) metre. The gravitational potential energy equals to (A) \(-\frac{3 G m^{2}}{a}\) (B) \(-\frac{3 G m}{a^{2}}\) (C) \(-\frac{3 G m}{a}\) (D) \(\frac{3 G m^{2}}{a}\)

If three uniform spheres, each having mass \(M\) and radius \(R\), are kept in such a way that each touches the other two, the magnitude of the gravitational force on any sphere due to the other two is (A) \(\frac{G M^{2}}{4 R^{2}}\) (B) \(\frac{2 G M^{2}}{R^{2}}\) (C) \(\frac{2 G M^{2}}{4 R^{2}}\) (D) \(\frac{\sqrt{3} G M^{2}}{4 R^{2}}\)

The period of revolution of planet \(A\) around the sun is 8 times that of \(B\). The distance of \(A\) from the sun is how many times greater than that of \(B\) from the sun? (A) 2 (B) 3 (C) 4 (D) 5

If the length of a simple pendulum is equal to the radius \(R\) of the earth, its time period will be (A) \(2 \pi \sqrt{R / g}\) (B) \(2 \pi \sqrt{R / 2 g}\) (C) \(2 \pi \sqrt{2 R / g}\) (D) \(\pi \sqrt{R / 2 g}\)

Given that mass of the earth is \(M\) and its radius is \(R\). A body is dropped from a height equal to the radius of the earth above the surface of earth. When it reaches the ground its velocity will be (A) \(\frac{G M}{R}\) (B) \(\left[\frac{G M}{R}\right]^{1 / 2}\) (C) \(\left[\frac{2 G M}{R}\right]^{1 / 2}\) (D) \(\left[\frac{2 G M}{R}\right]\)

The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration due to gravity. (A) Will be directed towards the centre but not the same everywhere. (B) Will have the same value everywhere but not directed towards the centre. (C) Will be same everywhere in magnitude directed towards the centre. (D) Cannot be zero at any point.

As observed from the earth, the sum appears to move in an approximate circular orbit. For the motion of another planet like mercury as observed from the earth, this would (A) be similarly true. (B) not be true because the force between the earth and mercury is not inverse square law. (C) not be true because the major gravitational force on mercury is due to the sun. (D) not be true because mercury is influenced by forces other than gravitational forces.

Different points in the earth are at slightly different distances from the sun and hence experience different forces due to gravitation. For a rigid body, we know that if various forces act at various points in it, the resultant motion is as if a net force acts on the CM (centre of mass) causing translation and a net torque at the CM causing rotation around an axis through the CM. For the earth-sun system (approximating the earth as a uniform density sphere). (A) The torque is zero. (B) The torque cause the earth to spin. (C) The rigid body result is not applicable since the earth is not even approximately a rigid body. (D) The torque causes the earth to move around the sun.

Satellites orbiting the earth have finite life and sometimes debris of satellites fall to the earth, This is because, (A) the solar cells and batteries in satellites run out. (B) the laws of gravitation predict a trajectory spiralling inwards. (C) of viscous forces causing the speed of satellite and hence height to gradually decrease. (D) of collisions with other satellites.

Both the earth and the moon are subject to the gravitational force of the sun. As observed from the sun, the orbit of the moon (A) will be elliptical. (B) will not be strictly elliptical because the total gravitational force on it is not central. (C) is not elliptical but will necessarily be a closed curve. (D) deviates considerably from being elliptical due to influence of planets other than the earth.

In the solar system, the inter-planetary region has chunks of matter (much smaller in size compared to planets) called asteroids. They (A) will not move around the sun, since they have very small masses compared to the sun. (B) will move in an irregular way because of their small masses and will drift away into outer space. (C) will move around the sun in closed orbits but not obey Kepler's laws. (D) will move in orbits like planets and obey Kepler's laws.

A particle initially at rest is displaced by applying a non-conservative force \(F\) in a uniform gravitational field. In the process, following physical quantities associated with the particle are measured. \(\Delta U=\) change in gravitational potential energy \(\Delta K=\) change in kinetic energy \(\Delta W_{1}=\) work done by the force \(F\) \(\Delta W_{2}=\) work done by the gravitational force (A) \(\Delta W_{2}=-\Delta U\) (B) \(\Delta K=\Delta W_{1}+\Delta W_{2}\) (C) \(\Delta K+\Delta U=\Delta W_{1}+\Delta W_{2}\) (D) \(\Delta W_{1}=\Delta W_{2}\)

Three planets of same density have radii \(R_{1}, R_{2}\) and \(R_{3}\) such that \(R_{1}=2 R_{2}=3 R_{3}\). The gravitational field at their respective surfaces are \(g_{1}, g_{2}\) and \(g_{3}\) and escape velocities from their surfaces are \(v_{1}, v_{2}\) and \(v_{3}\) respectively, then (A) \(g_{1} / g_{2}=2\) (B) \(g_{1} / g_{3}=3\) (C) \(v_{1} / v_{2}=1 / 4\) (D) \(v_{1} / v_{3}=3\)

Two spherical planets have the same mass but densities in the ratio \(1: 8\). For these planets, the (A) acceleration due to gravity will be in the ratio \(4: 1\). (B) acceleration due to gravity will be in the ratio \(1: 4\). (C) escape velocities from their surfaces will be in the ratio \(\sqrt{2}: 1\). (D) escape velocities from their surfaces will be in the ratio \(1: \sqrt{2}\).

Consider an attractive force which is central but is inversely proportional to the first power of distance. If such a particle is in circular orbit under such a force, which of the following statements are correct. (A) \(v \propto r\) (B) \(v \propto r^{\circ}\) (C) \(T \propto r\) (D) \(T \propto r^{\circ}\)

A solid sphere of uniform density and radius 4 units is located with its centre at the origin \(O\) of co-ordinates. Two spheres of equal radii 1 units, with their centres at \(A(-2,0,0)\) and \(B(2,0,0)\) respectively, are taken out of the solid leaving behind spherical cavities as shown in Fig. 7.12. Then (A) the gravitational field due to this object at the origin is zero. (B) the gravitational field at the point \(B(2,0,0)\) is zero. (C) the gravitational potential is same at all points on the circle \(y^{2}+z^{2}=36\) (D) the gravitational potential is same at all points on the circle \(y^{2}+z^{2}=4\)

Two objects of masses \(m\) and \(4 m\) are at rest at an infinite separation. They move towards each other under mutual gravitational force of attraction. If \(G\) is the universal gravitational constant. Then at separation \(r\) (A) the total energy of the two objects is zero. (B) their relative velocity of approach is \(\left(\frac{10 G m}{r}\right)^{\frac{1}{2}}\) in (C) the total kinetic energy of the objects is \(\frac{4 G m^{2}}{r}\). (D) net angular momentum of both the particles is zero about any point.

A sphere of density \(\rho\) and radius \(a\) has a concentric cavity of radius \(b\) as shown in the Fig. \(7.13\). Gravitation force \(F\) exerted by the sphere on the particle of mass \(m\), located at a distance \(r\) from the centre of sphere as a function \(r\) when \(b

A sphere of density \(\rho\) and radius \(a\) has a concentric cavity of radius \(b\) as shown in the Fig. \(7.13\). Gravitational potential energy as a function of \(r\), where \(r\) is the distance from the centre of the sphere. When \(0

In 1783, John Mitchell noted that if a body having same density as that of the sun but radius 500 times that of the sun, magnitude of its escape velocity will be greater than \(c\), the speed of light. All the light emitted by such a body will return to it. He, thus, suggested the existence of a black hole. \(v=c=\sqrt{\frac{2 G M}{R}}\) suggests that a body of mass \(M\) will act as a black hole if its radius \(R\) is less than or equal to a certain critical radius. Karl Schwarzchild, in 1926 , derived the expression for the critical radius \(R_{S}\) called Schwarzchild radius. The surface of the sphere with radius \(R_{S}\) surrounding a black hole is called event horizon. Schwarzchild radius \(R_{S}\) is (A) \(>\frac{2 G M}{c^{2}}\) (B) \(\frac{G M}{c^{2}}\) (C) \(<\frac{2 G M}{c^{2}}\) (D) \(\frac{2 G M}{c^{2}}\)

In 1783, John Mitchell noted that if a body having same density as that of the sun but radius 500 times that of the sun, magnitude of its escape velocity will be greater than \(c\), the speed of light. All the light emitted by such a body will return to it. He, thus, suggested the existence of a black hole. \(v=c=\sqrt{\frac{2 G M}{R}}\) suggests that a body of mass \(M\) will act as a black hole if its radius \(R\) is less than or equal to a certain critical radius. Karl Schwarzchild, in 1926 , derived the expression for the critical radius \(R_{S}\) called Schwarzchild radius. The surface of the sphere with radius \(R_{S}\) surrounding a black hole is called event horizon. Density of the sun is (A) \(14.1 \mathrm{~kg} \mathrm{~m}^{-3}\) (B) \(141.1 \mathrm{~kg} \mathrm{~m}^{-3}\) (C) \(1410 \mathrm{~kg} \mathrm{~m}^{-3}\) (D) None of these

In 1783, John Mitchell noted that if a body having same density as that of the sun but radius 500 times that of the sun, magnitude of its escape velocity will be greater than \(c\), the speed of light. All the light emitted by such a body will return to it. He, thus, suggested the existence of a black hole. \(v=c=\sqrt{\frac{2 G M}{R}}\) suggests that a body of mass \(M\) will act as a black hole if its radius \(R\) is less than or equal to a certain critical radius. Karl Schwarzchild, in 1926 , derived the expression for the critical radius \(R_{S}\) called Schwarzchild radius. The surface of the sphere with radius \(R_{S}\) surrounding a black hole is called event horizon. To make black hole with density of the sun, the ratio of radius of the object to that of sun should be (A) 5 (B) 50 (C) 500 (D) \(2.5\)

Suppose you are, as the header of a group of scientists working in NASA, sent to a planet named NSP2009 to study that planet you have following data with you. 1\. The planet NSP2009 is spherical in shape and of radius \(R\). 2\. The mass of the planet NSP2009 is M. If your team finds that the weight of any body inside the planet remains same as on its surface, what could be the possible value of mass density of the planet as a function of radial distance (where \(\rho_{0}\) is a content) \((R \geq>0)\) (A) \(\rho=\rho_{0} r^{0}\) (B) \(\rho=\rho_{0} r\) (C) \(\rho=\rho_{0} r^{-1}\) (D) \(\rho=\rho_{0} r^{-1 / 2}\)

A particle is projected vertically upwards from the surface of the earth with a kinetic energy equal to \(\frac{1}{3}\) times the minimum kinetic energy needed to escape. If radius of the earth is \(6400 \mathrm{~km}\), the maximum height attained by the particle (in \(\mathrm{km}\) ) from the surface of the earth is \(\frac{n R}{2}\) then the value of \(n\) is.

A particle of mass \(m\) is dropped from the earth surface into a tunnel dug through a diameter of the earth. The velocity with which it cross the centre of the earth is \(\sqrt{n g R}\), then the value of \(n\) is? Assume the earth to be of uniform density. Express your answer in terms of radius \(R\) of the earth and the acceleration due to gravity \(g\) at the surface of the earth.

Calculate the ratio \(m_{0} / m\) for a rocket if it is to escape from the earth. Given escape velocity \(=11.2 \mathrm{~km} / \mathrm{s}\) and exhaust speed of gases is \(2 \mathrm{~km} / \mathrm{s}\).

The kinetic energy needed to project a body of mass \(m\) from the earth's surface (radius \(R\) ) to infinity is (A) \(\frac{m g R}{2}\) (B) \(2 m g R\) (C) \(m g R\) (D) \(\frac{m g R}{4}\)

The escape velocity of a body depends upon mass as (A) \(m^{0}\) (B) \(m^{1}\) (C) \(m^{2}\) (D) \(m^{3}\)

Two spherical bodies of mass \(M\) and \(5 M\) and radii \(R\) and \(2 R\) respectively, are released in free space with initial separation between their centres equal to \(12 R\). If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is (A) \(2.5 R\) (B) \(4.5 R\) (C) \(7.5 R\) (D) \(1.5 R\)

The escape velocity for a body projected vertically upwards from the surface of earth is \(11 \mathrm{~km} / \mathrm{s}\). If the body is projected at an angle of \(45^{\circ}\) with the vertical, the escape velocity will be (A) \(11 \sqrt{2} \mathrm{~km} / \mathrm{s}\) (B) \(22 \mathrm{~km} / \mathrm{s}\) (C) \(11 \mathrm{~km} / \mathrm{s}\) (D) \(\frac{11}{\sqrt{2}} \mathrm{~m} / \mathrm{s}\)

A satellite of mass \(\mathrm{m}\) revolves around the earth of radius \(R\) at a height \(x\) from its surface. If \(g\) is the acceleration due to gravity on the surface of the earth, the orbital speed of the satellite is (A) \(g x\) (B) \(\frac{g R}{R-x}\) (C) \(\frac{g R^{2}}{R+x}\) (D) \(\left(\frac{g R^{2}}{R+x}\right)^{1 / 2}\)