Chapter 7

The time period of an earth satellite in circular orbit is independent of (A) the mass of the satellite. (B) radius of its orbit. (C) both the mass and radius of the orbit. (D) neither the mass of the satellite nor the radius of its orbit.

If \(g\) is the acceleration due to gravity on the earth's surface, the gain in the potential energy of an object of mass \(m\) raised from the surface of the earth to a height equal to the radius \(R\) of the earth, is (A) \(2 m g R\) (B) \(\frac{1}{2} m g R\) (C) \(\frac{1}{4} m g R\) (D) \(m g R\)

Suppose the gravitational force varies inversely as the \(n^{\text {th }}\) power of distance. Then the time period of a planet in circular orbit of radius \(R\) around the sun will be proportional to (A) \(R^{\left(\frac{n-2}{2}\right)}\) (B) \(R^{\left(\frac{n-1}{2}\right)}\) (C) \(R^{n}\) (D) \(R^{\left(\frac{n-2}{2}\right)}\)

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? (A) \(d=\frac{h}{2}\) (B) \(d=\frac{3 h}{2}\) (C) \(d=2 h\) (D) \(d=h\)

Average density of the earth (A) does not depend on \(g\). (B) is a complex function of \(g\). (C) is directly proportional to \(g\). (D) is inversely proportional to \(g\).

The change in the value of \(g\) at a height \(h\) above the surface of the earth is the same as that of 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? (A) \(d=\frac{h}{2}\) (B) \(d=\frac{3 h}{2}\) (C) \(d=2 h\) (D) \(d=h\)

A particle of mass \(10 \mathrm{~g}\) is kept on the surface of a uniform sphere of mass \(100 \mathrm{~kg}\) and radius \(10 \mathrm{~cm}\). Find the work to the done against be gravitational force between them, to take the particle far away from the sphere. (you may take \(\left.G=6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{Kg}^{-2}\right)\) (A) \(13.34 \times 10^{-10} \mathrm{~J}\) (B) \(3.33 \times 10^{-10} \mathrm{~J}\) (C) \(6.67 \times 10^{-9} \mathrm{~J}\) (D) \(6.67 \times 10^{-10} \mathrm{~J}\)

A planet in a distance 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) \(1.1 \mathrm{kms}^{-1}\) (B) \(11 \mathrm{kms}^{-1}\) (C) \(110 \mathrm{kms}^{-1}\) (D) \(0.11 \mathrm{kms}^{-1}\)

This question contains Statement 1 and Statement 2 of the four choices given after the statements, choose the one that best describes the two statements. [2008] Statement 1: For a mass \(M\) kept at the centre of a cube of side \(a\) the flux of gravitational field passing through its sides \(4 \pi G M\). Statement 2: If the direction of a field due to a point source is radial and its dependence on the distance \(r\) from the source is given as \(\frac{1}{r^{2}}\), its flux through a closed surface depends only on the strength of the source enclosed by the surface and not on the size or shape of the surface. (A) Statement 1 is false, Statement 2 is true. (B) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1 . (C) Statement 1 is false, Statement 2 is true; Statement 2 is not a correct explanation for Statement \(1 .\) (D) Statement 1 is true, Statement 2 is false.

The height at which the acceleration due to gravity becomes \(\frac{g}{9}\) (where \(g\) = the acceleration due to gravity on the surface of the earth) in terms of \(R\), the radius of the earth is (A) \(\frac{R}{\sqrt{2}}\) (B) \(R / 2\) (C) \(\sqrt{2} R\) (D) \(2 R\)

The mass of a 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) \(6.4 \times 10^{10} \mathrm{~J}\)

This question has Statement 1 and Statement \(2 .\) Of the four choices given after the statements, choose the one that best describes the two statements. Statement 1: Higher the range, greater is the resistance of ammeter. Statement 2: To increase the range of ammeter, additional shunt needs to be used across it. (A) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation of Statement 1 . (B) Statement 1 is true, Statement 2 is false. (C) Statement 1 is false, Statement 2 is true. (D) Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation of Statement 1 .

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 (A) \(\sqrt{\frac{G M}{R}}\) (B) \(\sqrt{2 \sqrt{2} \frac{G M}{R}}\) (C) \(\sqrt{\frac{G M}{R}(1+2 \sqrt{2})}\) (D) \(\frac{1}{2} \sqrt{\frac{G M}{R}(1+2 \sqrt{2})}\)

From a solid sphere of mass \(M\) and radius \(R\), a spherical portion of radius \(\frac{R}{2}\) is removed, as shown in Fig. 7.16. Taking Fig. \(7.16\) gravitational potential \(V=0\) at \(r=\infty\), the potential at the centre of the cavity thus formed is \((G=\) gravitational constant) \(\quad\) (A) \(\frac{-G M}{R}\) (B) \(\frac{-2 G M}{3 R}\) (C) \(\frac{-2 G M}{R}\) (D) \(\frac{-G M}{2 R}\)

The height at which the acceleration due to gravity becomes \(\frac{g}{9}\) (where \(g=\) the acceleration due to gravity on the surface of the earth) in terms of \(R\), the radius of the earth, is (A) \(2 R\) (B) \(\frac{R}{\sqrt{3}}\) (C) \(\frac{R}{2}\) (D) \(\sqrt{2} R\)

Two particle of equal mass \(m\) go around a circle of radius \(R\) under the action of their mutual gravitational attraction. The speed of each particle with respect to their centre of mass is (A) \(\sqrt{\frac{G m}{R}}\) (B) \(\sqrt{\frac{G m}{4 R}}\) (C) \(\sqrt{\frac{G M}{3 R}}\) (D) \(\sqrt{\frac{G m}{2 R}}\)

Two bodies of masses \(m\) and \(4 m\) are placed at a distance \(r .\) The gravitational potential at a point on the line joining them 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 mass of a 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) \(6.4 \times 10^{10} \mathrm{~J}\)

This question has Statement 1 and Statement 2. Of the four choices given after the statements, choose the one that best describes the two statements. Statement 1: Higher the range, greater is the resistance of ammeter. Statement 2: To increase the range of ammeter, additional shunt needs to be used across it. (A) Statement 1 is true, Statement 2 is true, Statement 2 is not the correct explanation of Statement 1 . (B) Statement 1 is true, Statement 2 is false. (C) Statement 1 is false, Statement 2 is true. (D) Statement 1 is true, Statement 2 is true, Statement 2 is correct explanation of Statement 1 .

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 (A) \(\sqrt{\frac{G M}{R}}\) (B) \(\sqrt{2 \sqrt{2} \frac{G M}{R}}\) (C) \(\sqrt{\frac{G M}{R}(1+2 \sqrt{2})}\) (D) \(\frac{1}{2} \sqrt{\frac{G M}{R}(1+2 \sqrt{2})}\)

From a solid sphere of mass \(M\) and radius \(R\), a spherical portion of radius \(\frac{R}{2}\) is removed, as shown in Fig. 7.17. Taking gravitational potential \(V=0\) at \(r=\infty\), the potential at the centre of the cavity thus formed is \((G=\) gravitational constant) (A) \(\frac{-G M}{R}\) (B) \(\frac{-2 G M}{3 R}\) (C) \(\frac{-2 G M}{R}\) (D) \(\frac{-G M}{2 R}\)

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) (A) \(\sqrt{g R}\) (B) \(\sqrt{g R / 2}\) (C) \(\sqrt{g R}(\sqrt{2}-1)\) (D) \(\sqrt{2 g R}\)