Chapter 10

If the earth shrinks such that its mass does not change but radius decreases to one-quarter of its original value, then one complete day will take (a) \(96 \mathrm{~h}\) (b) \(48 \mathrm{~h}\) (c) \(6 \mathrm{~h}\) (d) \(1.5 \mathrm{~h}\)

The gravitational field due to a mass distribution is \(1=\frac{C}{x^{2}}\) in \(x\) direction. Here \(C\) is constant. Taking the gravitational potential to be zero at infinity, potential at \(x\) is (a) \(\frac{2 C}{x}\) (b) \(\frac{C}{x}\) (c) \(\frac{2 C}{x^{2}}\) (d) \(\frac{C}{2 x^{2}}\)

Here, \(m=20 \mathrm{~kg}, l=1 \mathrm{~m}, r=0.2 \mathrm{~m}\) Moment of inertia about its geometrical axis is $$ \begin{aligned} I &=\frac{1}{2} m r^{2} \\ &=\frac{1}{2} \times 20(0.2)^{2}=0.4 \mathrm{~kg} \mathrm{~m}^{2} \end{aligned} $$

A space ship moves from earth to moon and back. Th. greatest energy required for the space ship is \(t\) overcome the difficulty in (a) entering the earth's gravitational field (b) take off from earth's field (c) take off from lunar surface (d) entering the moon's lunar surface

Infinite number of masses, each of \(1 \mathrm{~kg}\) are placed along the \(x\)-axis at \(x=+1 m, \pm 2 m_{1}, \pm 4 m\), \(\pm 8 m, \pm 16 m \ldots\) The magnitude of the resultant gravitational potential in terms of gravitational constant \(G\) at the origin \((x=0)\) is (a) \(\mathrm{G} / 2\) (b) \(\underline{G}\) (c) \(2 \mathrm{G}\) (d) \(4 G\)

The change in potential energy when a body of mass \(m\) is raised to a height \(n R\) from the centre of earth ( \(R=\) radius of earth) (a) \(m g R \frac{(n-1)}{n}\) (b) \(n m g R\) (c) \(m g R \frac{n^{2}}{n^{2}+1}\) (d) \(m g R \frac{n}{n+1}\)

A planet in a distant solar system is 10 times more massive than the earth and its radius is 10 times smaller. Given that the escape velocity from the earth is \(11 \mathrm{kms}^{-1}\), the escape velocity from the surface of the planet would be (a) \(0.11 \mathrm{kms}^{-1}\) (b) \(1.1 \mathrm{kms}^{-1}\) (c) \(11 \mathrm{kms}^{-1}\) (d) \(110 \mathrm{kms}^{-1}\)

The escape velocity of a body on the surface of earth is \(11.2 \mathrm{kms}^{-1}\). If the mass of the earth is doubled and its radius halved, the escape velocity becomes (a) \(5.6 \mathrm{kms}^{-1}\) (b) \(11.2 \mathrm{kms}^{-1}\) (c) \(22.4 \mathrm{kms}^{-1}\) (d) \(44.8 \mathrm{kms}^{-1}\)

The gravitational field in a region is given by \(\mathbf{I}=(4 \hat{\mathbf{i}}+\hat{\mathbf{j}}) \mathbf{N k g}^{-1}\). Work done by this field is zero when a particle is moved along the line (a) \(x+y=6\) (b) \(x+4 y=6\) (c) \(y+4 x=6\) (d) \(x-y=6\)

The change in potential energy when a body of mass \(m\) is raised to a height \(n R\) from earth's surface is \((R=\) radius of the earth) (a) \(m g R \frac{n}{(n-1)}\) (b) \(m g R\) (c) \(m g R \frac{n}{(n+1)}\) (d) \(m g R \frac{n^{2}}{\left(n^{2}+1\right)}\)

A satellite orbits the earth at a height of \(400 \mathrm{~km}\) above the surface. How much energy must be expended to rocket the satellite out of the earth's gravitational influence? Mass of the satellite \(=200 \mathrm{~kg}\), mass of the earth \(=6.0 \times 10^{24} \mathrm{~kg}\), radius of the earth \(=6.4 \times 10^{6} \mathrm{~m}, G=6.67 \times 10^{-11} \mathrm{~N}-\mathrm{m}^{2} / \mathrm{kg}^{2}\). (a) \(5.2 \times 10^{10} \mathrm{~J}\) (b) \(3 \times 10^{6} \mathrm{~J}\) (c) \(4 \times 10^{6} \mathrm{~J}\) (d) \(6 \times 10^{9} \mathrm{~J}\)

If \(g_{E}\) and \(g_{M}\) are the acceleration due to gravity on the surfaces of the earth and the moon respectively and if Millikan's oil drop experiment could be performed on two surfaces, one will find the ratio electronic charge on the moon electronic charge on the earth to be (a) \(g_{m} / g_{E}\) (b) 1 (c) 0 (d) \(g_{E} / g_{M}\)

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

The escape velocity on the surface of earth is \(11.2 \mathrm{kms}^{-1} .\) If mass and radius of a planet is 4 and 2 times respectively, than that of earth, the escape velocity on the planet (a) \(11.2 \mathrm{kms}^{-1}\) (b) \(1.12 \mathrm{kms}^{-1}\) (c) \(22.4 \mathrm{kms}^{-1}\) (d) \(15.8 \mathrm{kms}^{-1}\)

A simple pendulum has a time period \(T_{1}\) when on the earth's surface and \(T_{2}\) when taken to a height \(2 R\) above the earth's surface when \(R\) is \(2 R\) above earth's surface where \(R\) is the radius of the earth. The value of \(\left(T_{1} / T_{2}\right)\) is (a) \(1 / 9\) (b) \(1 / 3\) (c) \(\sqrt{3}\) (d) 9

The mass of a planet is six times that of the earth. The radius of the planet is twice that of the earth. If the escape velocity from the earth is \(v\), then the escape velocity from the planet is (a) \(\sqrt{3} v_{e}\) (b) \(\sqrt{2} v_{e}\) (c) \(v_{e}\) (d) \(\sqrt{5} v_{e}\)

Out of the following, the only correct statement about satellites is (a) A satellite cannot move in a stable orbit in a plane passing through the earth's centre (b) Geostationary satellites are launched in the equatorial plane (c) We can use just one geostationary satellite for global communication around the globe (d) The speed of satellite increases with an increase in the radius of its orbit

A satellite in a circular orbit of radius \(R\) has a period of \(4 \mathrm{~h}\). Another satellite with orbital radius \(3 R\) around the same planet will have a period (in hours) (a) 16 (b) 4 (c) \(4 \sqrt{27}\) (d) \(4 \sqrt{8}\)

A satellite \(S\) is moving in an elliptical orbit around earth. The mass of the satellite is very small compared to the mass of the earth? (a) The acceleration of \(S\) is always directed towards the centre of the earth (b) The angular momentum of \(S\) about the centre of the earth changes in direction but its magnitude remains constant (c) The total mechanical energy of \(S\) varies periodically with time (d) The linear momentum of \(S\) remains constant in magnitude

If earth suddenly shrinks by one-third of its present radius, the acceleration due to gravity will be (a) \(\frac{2}{3} g\) (b) \(\frac{3}{2} g\) (c) \(\frac{4}{9} g\) (d) \(\frac{9}{4} g\)

A satellite is placed in a circular orbit around earth at such a height that it always remains stationary with respect to earth surface. In such case, its height from the earth surface is (a) \(32000 \mathrm{~km}\) (b) \(36000 \mathrm{~km}\) (c) \(6400 \mathrm{~km}\) (d) \(4800 \mathrm{~km}\)

The escape velocity for the earth is \(v_{\mathrm{es}}(s) .\) The escape velocity for a planet whose radius is four times and density is nine times that of the earth is (a) \(36 v_{\text {es(e) }}\) (b) \(12 v_{\text {es }(e)}\) (c) \(6 v_{\text {es }(e)}\) (d) \(20 v_{\text {es(e })}\)

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

Two bodies of masses \(m\) and \(4 m\) are placed at a distance \(r .\) The gravitational potential at a point on the line joining, then where the gravitational field is zero, is (a) \(\frac{-4 G M}{r}\) (b) \(\frac{-6 G M}{r}\) (c) \(\frac{-9 G M}{r}\) (d) zero

The orbital velocity of an artificial satellite in a circular orbit just above the earth's surface is \(v\). For a satellite orbiting at an altitude of half of the earth's radius, the orbital velocity is (a) \(\frac{3}{2} v\) (b) \(\sqrt{\frac{3}{2}} v\) (c) \(\sqrt{\frac{2}{3}} v\) (d) \(\frac{2}{3} v\)

The mass of spaceship is \(1000 \mathrm{~kg}\). It is to be launched from the earth's surface out into free space. The value of \(g\) and \(R\). (radius of earth) are \(10 \mathrm{~m} / \mathrm{s}^{2}\) and \(6400 \mathrm{~km}\) respectively. The required energy for this work will be (a) \(6.4 \times 10^{11} \mathrm{~J}\) (b) \(6.4 \times 10^{8} \mathrm{~J}\) (c) \(6.4 \times 10^{9} \mathrm{~J}\) (d) \(6.4 \times 10^{10} \mathrm{~J}\)

If \(v_{e}\) and \(v_{o}\) represent the escape velocity and orbital velocity of satellite corresponding to a circular orbit of radius \(R\), then (a) \(v_{e}=v_{o}\) (b) \(\sqrt{2} v_{o}=v_{e}\) (c) \(v_{e}=\frac{v_{o}}{\sqrt{2}}\) (d) \(v_{e}\) and \(v_{o}\) are not related

If the moon is to escape from the gravitational field of the earth forever, it will require a velocity (a) \(11.2 \mathrm{kms}^{-1}\) (b) less than \(11.2 \mathrm{kms}^{-1}\) (c) slightly more than \(11.2 \mathrm{kms}^{-1}\) (d) \(22.4 \mathrm{kms}^{-1}\)

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 \(2 R ? (a) \)\frac{5 \mathrm{GmM}}{6 R}\( (b) \)\frac{2 G m M}{3 R}\( (c) \)\frac{G m M}{2 R}\( (d) \)\frac{G m M}{3 R}$

The escape velocity from the earth is \(11 \mathrm{kms}^{-1}\). The escape velocity from a planet having twice the radius and the same mean density as the earth would be (a) \(5.5 \mathrm{kms}^{-1}\) (b) \(11 \mathrm{kms}^{-1}\) (c) \(15.5 \mathrm{kms}^{-1}\) (d) \(22 \mathrm{kms}^{-1}\)

The escape velocity for a body projected vertically| upwards from the surface of the earth is \(11.2 \mathrm{kms}^{-1}\) If the body is projected in a direction making an angle of \(45^{\circ}\) with the vertical, the escape velocity will be (a) \(11.2 \mathrm{kms}^{-1}\) (b) \(11.2 \times \sqrt{2} \mathrm{kms}^{-1}\) (c) \(11.2 \times 2 \mathrm{kms}^{-1}\) [d) \(11.2 / \sqrt{2} \mathrm{kms}^{-1}\)

The ratio of the radii of the planets \(P_{1}\) and \(P_{2}\) is \(a\). The ratio of their acceleration due to gravity is \(b .\) The ratio of the escape velocities from them will be (a) \(a b\) (b) \(\sqrt{a b}\) (c) \(\sqrt{a / b}\) (d) \(\sqrt{b / a}\)

The mass of the moon is \(1 / 81\) of earth's mass and its radius \(1 / 4\) th that of the earth. If the escape velocity from the earth's surface is \(11.2 \mathrm{kms}^{-1}\), its value for the moon will be (a) \(0.15 \mathrm{kms}^{-1}\) (b) \(5 \mathrm{kms}^{-1}\) (c) \(2.5 \mathrm{kms}^{-1}\) (d) \(0.5 \mathrm{kms}^{-1}\)

The period of revolution of planet \(A\) around the sun is 8 times that \(B\). 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

The largest and the shortest distance of the earth from the sun are \(r_{1}\) and \(r_{2}\), its distance from the sun when it is perpendicular to the major axis of the orbit drawn from the sun, is (a) \(\frac{r_{1}+r_{2}}{4}\) (b) \(\frac{r_{1} r_{2}}{r_{1}+r_{2}}\) (c) \(\frac{2 r_{1} r_{2}}{r_{1}+\hbar_{2}}\) (d) \(\frac{r_{1}+r_{2}}{3}\)

In our solar system, the inter-planetary region has chunks of matter (much smaller in size compared to planets) called asteroids. They [NCERT Exemplar] (a) will not move around the sun since they have very small masses compared to 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 comet of mass \(m\) moves in a highly elliptical orbit around the sun of mass \(M\). The maximum and minimum distances of the comet from the centre of the sun are \(r_{1}\) and \(r_{2}\) respectively. The magnitude of angular momentum of the comet with respect to the centre of sun is (a) \(\left[\frac{G M r_{1}}{\left(r_{1}+r_{2}\right)}\right]^{1 / 2}\) (b) \(\left[\frac{G M m r_{1}}{\left(r_{1}+r_{2}\right)}\right]^{1 / 2}\) (c) \(\left(\frac{2 G m^{2} r_{12}}{r_{1}+r_{2}}\right)^{1 / 2}\) (d) \(\left(\frac{2 G M m^{2} r_{12}}{\left(r_{1}+r_{2}\right)}\right)^{1 / 2}\)