Chapter 6

Two identical solid copper spheres of radius \(R\) are placed in contact with each other. The gravitational force between them is proportional to (A) \(\mathrm{R}^{2}\) (B) \(\mathrm{R}^{-2}\) (C) \(\mathrm{R}^{-4}\) (D) \(\mathrm{R}^{4}\)

The atmosphere is held to the earth by (A) clouds (B) Gravity (C) Winds (D) None of the above

Two sphere of mass \(\mathrm{m}_{1}\) and \(\mathrm{m}_{2}\) are situated in air and the gravitational force between them is \(F\). The space around the masses is now filled with liquid of specific gravity 3 . The gravitational force will now be (A) \(\mathrm{F}\) (B) \(3 \mathrm{~F}\) (C) \(\mathrm{F} / 3\) (D) \(\mathrm{F} / 9\).

A satellite of the earth is revolving in a circular orbit with a uniform speed \(\mathrm{v} .\) If the gravitational force suddenly disappears, the satellite will (A) Continue to move with velocity \(\mathrm{v}\) along the original orbit. (B) Move with a Velocity \(\mathrm{v}\), tangentially to the original orbit. (C) Fall down with increasing velocity. (D) Ultimately come to rest somewhere on the original orbit.

Correct form of gravitational law is (A) \(\mathrm{F}=-\left[\left(\mathrm{Gm}_{1} \mathrm{~m}_{2}\right) / \mathrm{r}^{2}\right]\) (B) \(\mathrm{F}^{-}=-\left[\left(\mathrm{Gm}_{1} \mathrm{~m}_{2}\right) / \mathrm{r}^{2}\right]\) (C) \(\mathrm{F}^{-}=-\left[\left(\mathrm{Gm}_{1} \mathrm{~m}_{2}\right) / \mathrm{r}^{2}\right] \hat{\mathrm{r}}\) (B) \(\mathrm{F}^{\rightarrow}=-\left[\left(\mathrm{Gm}_{1} \mathrm{~m}_{2}\right) / \mathrm{r}^{3}\right] \mathrm{r}^{-}\)

Mass \(M\) is divided into two parts \(\mathrm{xM}\) and \((1-\mathrm{x}) \mathrm{M}\). For a given separation, the value of \(\mathrm{x}\) for which the gravitational force between the two pieces becomes maximum is (A) 1 (B) 2 (C) \(1 / 2\) (D) \(4 / 5\)

The earth (mass \(=6 \times 10^{24} \mathrm{~kg}\) ) revolves around the sun with angular velocity \(2 \times 10^{-7} \mathrm{rad} / \mathrm{sec}\) in a circular orbit of radius \(1.5 \times 10^{8} \mathrm{~km} .\) The force exerted by the sun on the earth is \(=\ldots \ldots \ldots \ldots . \mathrm{N}\) (A) \(18 \times 10^{25}\) (b) zero (C) \(27 \times 10^{39}\) (D) \(36 \times 10^{21}\)

The distance of the moon and earth is \(D\) the mass of earth is 81 times the mass of moon. At what distance from the center of the earth, the gravitational force will be zero (A) \(\mathrm{D} / 2\) (B) \([(12 \mathrm{D}) / 3]\) (C) \((4 \mathrm{D} / 3)\) (D) \((9 \mathrm{D} / 10)\)

Three equal masses of \(\mathrm{m} \mathrm{kg}\) each are placed at the vertices of an equilateral triangle \(\mathrm{PQR}\) and a mass of \(2 \mathrm{~m} \mathrm{~kg}\) is placed at the centroid 0 of the triangle which is at a distance of \(\sqrt{2} \mathrm{~m}\) from each of vertices of triangle. The force in newton acting on the mass \(2 \mathrm{~m}\) is \(=\ldots \ldots \ldots\).. (A) 2 (B) 1 (C) \(\sqrt{2}\) (D) zero

Two point masses \(\mathrm{A}\) and \(\mathrm{B}\) having masses in the ratio \(4: 3\) are separated by a distance of \(\operatorname{lm}\). When another point mass of mass \(\mathrm{M}\) is placed in between \(\mathrm{A}\) and \(\mathrm{B}\) the forces \(\mathrm{A}\) and is \((1 / 3 \mathrm{rd})\) of the force between \(\mathrm{B}\) and \(\mathrm{C}\), Then the distance \(\mathrm{C}\) from \(\mathrm{A}\) is \(=\ldots \ldots \ldots \mathrm{m}\) (A) \((2 / 3)\) (B) \(1 / 3\) (C) \(1 / 4\) (D) \(2 / 7\)

As we go from the equator to the poles, the value of \(g \ldots \ldots \ldots\) (A) Remains constant (B) Decreases (C) Increases (D) Decreases upto latitude of \(45^{\circ}\)

If \(R\) is the radius of the earth and \(g\) the acceleration due to gravity on the earth's surface, the mean density of the earth is \(=\ldots \ldots \ldots\) (A) \([(4 \pi \mathrm{G}) /(3 \mathrm{~g} \mathrm{R})]\) (B) \([(3 \pi R) /(4 \mathrm{gG})]\) (C) \([(3 \mathrm{~g}) /(4 \pi \mathrm{RG})]\) (D) \([(\pi R G) /(12 g)]\)

The radius of the earth is \(6400 \mathrm{~km}\) and \(\mathrm{g}=10 \mathrm{~ms}^{-2} .\) In order that a body of \(5 \mathrm{~kg}\) weights zero at the equator, the angular speed of the earth is \(=\ldots \ldots \ldots \mathrm{rad} / \mathrm{sec}\) (A) \((1 / 80)\) (B) \([1 /(400)]\) (C) \([1 /(800)]\) (D) \([1 /(600)]\)

The time period of a simple pendulum on a freely moving artificial satellite is .......... sec (A) 0 (B) 2 (C) 3 (D) Infinite

A spherical planet far out in space has mass \(\mathrm{M}_{0}\) and diameter \(\mathrm{D}_{0}\). A particle of \(\mathrm{m}\) falling near the surface of this planet will experience an acceleration due to gravity which is equal to (A) \(\left[\left(\mathrm{GM}_{0}\right) /\left(\mathrm{D}_{\circ}^{2}\right)\right]\) (B) \(\left[\left(4 \mathrm{mGM}_{0}\right) /\left(\mathrm{D}_{0}^{2}\right)\right]\) (C) \(\left[\left(4 \mathrm{GM}_{0}\right) /\left(\mathrm{D}_{0}^{2}\right)\right]\) (D) \(\left[\left(\mathrm{GmM}_{0}\right) /\left(\mathrm{D}_{\circ}^{2}\right)\right]\)

A body weights \(700 \mathrm{~g} \mathrm{wt}\) on the surface of earth. How much it weight on the surface of planet whose mass is \(1 / 7\) and radius is half that of the earth (A) \(200 \mathrm{~g} \mathrm{wt}\) (B) \(400 \mathrm{~g} \mathrm{wt}\) (C) \(50 \mathrm{~g} \mathrm{wt}\) (D) \(300 \mathrm{~g}\) wt.

The moon's radius is \(1 / 4\) that of earth and its mass is \(1 / 80\) times that of the earth. If g represents the acceleration due to gravity on the surface of earth, that on the surface of the moon is (A) \(g / 4\) (B) \(\mathrm{g} / 5\) (c) \(\mathrm{g} / 6\) (D) \(\mathrm{g} / 8\)

If the density of small planet is that of the same as that of the earth while the radius of the planet is \(0.2\) times that of the earth, the gravitational acceleration on the surface of the planet is (A) \(0.2 \mathrm{~g}\) (B) \(0.4 \mathrm{~g}\) (C) \(2 \mathrm{~g}\) (D) \(4 \mathrm{~g}\)

If mass of a body is \(\mathrm{M}\) on the earth surface, than the mass of the same body on the moon surface is (A) \(\mathrm{M} / 6\) (B) 56 (C) \(\mathrm{M}\) (D) None of these

If the radius of earth is \(\mathrm{R}\) then height \({ }^{\prime} \mathrm{h}\) ' at which value of ' \(\mathrm{g}\) ' becomes one-fourth is (A) \(\mathrm{R} / 4\) (B) \(3 \mathrm{R} / 4\) (C) \(\mathrm{R}\) (D) \(\mathrm{R} / 8\)

If the mass of earth is 80 times of that of a planet and diameter is double that of planet and ' \(\mathrm{g}\) ' on the earth is \(9.8 \mathrm{~ms}^{-2}\), then the value of \(\mathrm{g}^{\prime}\) on that planet is \(=\ldots \ldots \ldots \mathrm{ms}^{-2}\) (A) \(4.9\) (B) \(0.98\) (C) \(0.49\) (D) 49

A body weight \(500 \mathrm{~N}\) on the surface of the earth. How much would it weight half way below the surface of earth (A) \(125 \mathrm{~N}\) (B) \(250 \mathrm{~N}\) (C) \(500 \mathrm{~N}\) (D) \(1000 \mathrm{~N}\)

The radii of two planets are respectively \(\mathrm{R}_{1}\) and \(\mathrm{R}_{2}\) and their densities are respectively \(\rho_{1}\) and \(\rho_{2}\) the ratio of the accelerations due to gravity at their surface is (A) \(g_{1}: g_{2}=\left(\rho_{1} / R_{1}^{2}\right) \cdot\left(\rho_{2} / R_{2}^{2}\right)\) (B) \(\mathrm{g}_{1}: \mathrm{g}_{2}=\mathrm{R}_{1} \mathrm{R}_{2}: \rho_{1} \rho_{2}\) (C) \(g_{1}: g_{2}=R_{1} \rho_{2} \cdot R_{2} p_{1}\) (D) \(g_{1}: g_{2}=R_{1} \rho_{1}: R_{2} \rho_{2}\)

At what height over the earth's pole, the free fall acceleration decreases by one percent \(=\ldots \ldots \ldots \mathrm{km}(\mathrm{Re}=6400 \mathrm{~km})\) (A) 32 (B) 80 (C) \(1.253\) (D) 64

Weight of a body is maximum at (A) moon (B) poles of earth (C) Equator of earth (D) Center of earth

At what distance from the center of earth, the value of acceleration due to gravity \(g\) will be half that of the surfaces \((\mathrm{R}=\) Radius of earth \()\) (A) \(2 \mathrm{R}\) (B) \(\mathrm{R}\) (C) \(1.414 \mathrm{R}\) (D) \(0.414 \mathrm{R}\)

The acceleration due to gravity near the surface of a planet of radius \(\mathrm{R}\) and density \(\mathrm{d}\) is proportional to (A) \(\mathrm{d} / \mathrm{R}^{2}\) (B) \(\mathrm{d} \mathrm{R}^{2}\) (C) \(\mathrm{dR}\) (D) \(\mathrm{d} / \mathrm{R}\)

The density of a newly discovered planet is twice that of earth. The acceleration due to gravity at the surface of the planet is equal to that at the surface of earth. If the radius of the earth is \(\mathrm{R}\), the radius of planet would be (A) \(2 \mathrm{R}\) (B) \(4 \mathrm{R}\) (C) \(1 / 4 \mathrm{R}\) (D) \(\mathrm{R} / 2\)

Density of the earth is doubled keeping its radius constant then acceleration, due to gravity will be \(-m s^{-2}\) \(\left(\mathrm{g}=9.8 \mathrm{~ms}^{2}\right)\) (A) \(19.6\) (B) \(9.8\) (C) \(4.9\) (D) \(2.45\)

If the value of ' \(\mathrm{g}\) ' acceleration due to gravity, at earth surface is \(10 \mathrm{~ms}^{-2}\). its value in \(\mathrm{ms}^{-2}\) at the center of earth, which is assumed to be a sphere of Radius ' \(\mathrm{R}\) 'meter and uniform density is (A) 5 (B) \(10 / \mathrm{R}\) (C) \(10 / 2 \mathrm{R}\) (D) zero

The acceleration of a body due to the attraction of the earth (radius R) at a distance \(2 \mathrm{R}\) from the surface of the earth is \(=\) (g \(=\overline{\text { acceleration due to gravity at the surface of earth })}\) (A) \(\mathrm{g} / 9\) (B) \(\mathrm{g} / 3\) (C) \(\mathrm{g} / 4\) (D) 9

The height at which the weight of a body becomes \(1 / 16\) th its weight on the surface of (radius \(\mathrm{R}\) ) is (A) \(3 \mathrm{R}\) (B) \(4 \mathrm{R}\) (C) \(5 \mathrm{R}\) (D) \(15 \mathrm{R}\)

In a gravitational field, at a point where the gravitational potential is zero (A) The gravitational field is necessarily zero (B) The gravitational field is not necessarily zero (C) Nothing can be said definitely, about the gravitational field (D) None of these

What is the intensity of gravitational field at the center of spherical shell (A) \(\left(\mathrm{Gm} / \mathrm{r}^{2}\right)\) (B) \(\mathrm{g}\) (C) zero (D) None of these

Escape velocity of a body of \(1 \mathrm{~kg}\) on a planet is \(100 \mathrm{~ms}^{-1}\). Gravitational potential energy of the body at the planet is \(=\) \(\begin{array}{ll}\text { (A) } \overline{-5000} & \text { (B) }-1000\end{array}\) (C) \(-2400\) (D) 5000

A body of mass \(\mathrm{m} \mathrm{kg}\) starts falling from a point \(2 \mathrm{R}\) above the earth's surface. Its \(\mathrm{K} . \mathrm{E}\). when it has fallen to a point ' \(\mathrm{R}\) ' above the Earth's surface \(=\ldots \ldots \ldots \ldots J\) [R - Radius of Earth, M-mass of Earth G-Gravitational constant \(]\) (A) \((1 / 2)[(\mathrm{GMm}) / \mathrm{R}]\) (B) \((1 / 6)[(\mathrm{GMm}) / \mathrm{R}]\) (C) \((2 / 3)[(\mathrm{GMm}) / \mathrm{R}]\) (D) \((1 / 3)[(\mathrm{GMm}) / \mathrm{R}]\)

The Gravitational P.E. of a body of mass \(\mathrm{m}\) at the earth's surface is \(-\mathrm{mgRe}\). Its gravitational potential energy at a height \(\operatorname{Re}\) from the earth's surface will be \(=\ldots \ldots \ldots\) here (Re is the radius of the earth) (A) \(-2 \mathrm{mgRe}\) (B) \(2 \mathrm{mgRe}\) (C) \((1 / 2) \mathrm{mg} \mathrm{Re}\) (D) \(-(1 / 2) \mathrm{mg} \operatorname{Re}\)

A body is projected vertically upwards from the surtace of a planet of radius \(\mathrm{R}\) with a velocity equal to half the escape velocity for that planet. The maximum height attained by the body is \(\ldots \ldots \ldots \ldots\) (A) \((\mathrm{R} / 3)\) (B) \((\mathrm{R} / 2)\) (C) \((\mathrm{R} / 4)\) (D) \((\mathrm{R} / 5)\)

Energy required to move a body of mass \(\mathrm{m}\) from from an orbit of radius \(2 \mathrm{R}\) to \(3 \mathrm{R}\) is \(\ldots \ldots \ldots \ldots\) (A) \(\left[(\mathrm{GMm}) /\left(12 \mathrm{R}^{2}\right)\right]\) (B) \(\left[(\mathrm{GMm}) /\left(3 \mathrm{R}^{2}\right)\right]\) (C) \([(\mathrm{GMm}) /(8 \mathrm{R})]\) (D) \([(\mathrm{GMm}) /(6 \mathrm{R})]\)

Radius of orbit of satellite of earth is \(\mathrm{R}\). Its \(\mathrm{KE}\) is proportional to (A) \((1 / R)\) (B) \((1 / \sqrt{\mathrm{R}})\) (C) \(\mathrm{R}\) (D) \(\left(1 / \mathrm{R}^{3 / 2}\right)\)

The escape velocity for a sphere of mass \(\mathrm{m}\) from earth having mass \(\mathrm{M}\) and Radius \(\mathrm{R}\) mass is given by (A) \(\sqrt{[}(2 \mathrm{GM}) / \mathrm{R}]\) (B) \(2 \sqrt{(\mathrm{GM} / \mathrm{R})}\) (C) \(\sqrt{[}(2 \mathrm{GMm}) / \mathrm{R}]\) (D) \(\sqrt{(\mathrm{GM} / \mathrm{R})}\)

The escape velocity for a rocket from earth is \(11.2 \mathrm{kms}^{-1}\) value on a planet where acceleration due to gravity is double that on earth and diameter of the planet is twice that of earth will be \(=\ldots \ldots \ldots \ldots \mathrm{kms}^{-1}\) (A) \(11.2\) (B) \(22.4\) (C) \(5.6\) (C) \(53.6\)

The escape velocity from the earth is about \(11 \mathrm{kms}^{-1}\). The escape velocity from a planet having twice the radius and the same mean density as the earth is \(=\ldots \ldots \ldots \ldots \mathrm{kms}^{-1}\). (A) 22 (B) 11 (C) \(5.5\) (D) \(15.5\)

If \(\mathrm{g}\) is the acceleration due to gravity at the earth's surface and \(\mathrm{r}\) is the radius of the earth, the escape velocity for the body to escape out of earth's gravitational field is \(\ldots \ldots \ldots\) (A) \(\mathrm{gr}\) (B) \(\sqrt{(2 \mathrm{gr})}\) (C) \(\mathrm{g} / \mathrm{r}\) (D) \(\mathrm{r} / \mathrm{g}\)

The escape velocity of a projectile from the earth is approximately (A) \(11.2 \mathrm{kms}^{-1}\) (B) \(112 \mathrm{kms}^{-1}\) (C) \(11.2 \mathrm{~ms}^{-1}\) (D) \(1120 \mathrm{kms}^{-1}\)

The escape velocity of an object from the earth depends upon the mass of earth (M), its mean density ( \(p\) ), its radius (R) and gravitational constant (G), thus the formula for escape velocity is (A) \(U=\mathrm{R} \sqrt{[}(8 \pi / 3) \mathrm{Gp}]\) (C) \(\mathrm{U}=\sqrt{(2 \mathrm{GMR})}\) (D) \(U=\sqrt{\left[(2 \mathrm{GMR}) / \mathrm{R}^{2}\right]}\)

Two small and heavy sphere, each of mass \(\mathrm{M}\), are placed distance r apart on a horizontal surface the gravitational potential at a mid point on the line joining the center of spheres is (A) zero (B) \(-(\mathrm{GM} / \mathrm{r})\) (C) \(-[(2 \mathrm{GM}) / \mathrm{r}]\) (D) \(-[(4 \mathrm{GM}) / \mathrm{r}]\)

The escape velocity of a body from earth's surface is Ve. The escape velocity of the same body from a height equal to 7 R from earth's surface will be (A) \((\mathrm{Ve} / \sqrt{2})\) (B) \((\mathrm{Ve} / 2)\) (C) \((\mathrm{Ve} / 2 \sqrt{2})\) (D) \((\mathrm{Ve} / 4)\)

The escape velocity of a planet having mass 6 times and radius 2 times as that of earth is (A) \(\sqrt{3} \mathrm{~V}_{\mathrm{e}}\) (B) \(3 \mathrm{~V}_{\mathrm{e}}\) (C) \(\sqrt{2} \mathrm{~V}_{\mathrm{e}}\) (D) \(2 \mathrm{~V}_{\mathrm{e}}\)

There are two planets, the ratio of radius of two planets is \(\mathrm{k}\) but the acceleration due to gravity of both planets are \(\mathrm{g}\) what will be the ratio of their escape velocity. (A) \((\mathrm{kg})^{1 / 2}\) (B) \((\mathrm{kg})^{-1 / 2}\) (C) \((\mathrm{kg})^{2}\) (D) \((\mathrm{kg})^{-2}\)