A particle of mass \(\mathrm{m}\) is thrown upwards from the surface of the earth, with a velocity \(\mathrm{u}\). The mass and the radius of the earth are, respectively, \(\mathrm{M}\) and \(\mathrm{R}\). \(\mathrm{G}\) is the gravitational constant and \(\mathrm{g}\) is the acceleration due to gravity on the surface of the earth. The minimum value of \(\mathrm{u}\) so that the particle does not return back to earth is:
1. \(\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}^2}} \)
2. \(\sqrt{\frac{2 \mathrm{GM}}{\mathrm{R}}} \)
3.\(\sqrt{\frac{2 \mathrm{gM}}{\mathrm{R}^2}} \)
4. \(\sqrt{ \mathrm{2gR^2}}\)
The dependence of acceleration due to gravity 'g' on the distance 'r' from the centre of the earth, assumed to be a sphere of radius R of uniform density, is as shown in figure below:
The correct figure is:
1. a
2. b
3. c
4. d
The additional kinetic energy to be provided to a satellite of mass \(m\) revolving around a planet of mass \(M,\) to transfer it from a circular orbit of radius \(R_1\) to another of radius \(R_2\) (\(R_2>R_1\)) is:
1. \(GmM\)
2. \(2GmM\)
3.
4. \(GmM\)
Which one of the following plots represents the variation of a gravitational field on a particle with distance \(r\) due to a thin spherical shell of radius \(R?\)
(\(r\) is measured from the centre of the spherical shell)
1. | 2. | ||
3. | 4. |
1. | \(v_o=v_e\) | 2. | \(v_e=\sqrt{2v_o}\) |
3. | \(v_e=\sqrt{2}~v_o\) | 4. | \(v_o=\sqrt{2}~v_e\) |
1. | \(\left(\frac{{GM}}{2 {R}}\right)^{\frac{1}{2}} \) | 2. | \(\left(\frac{{g} R}{4}\right)^{\frac{1}{2}} \) |
3. | \( \left(\frac{2 g}{R}\right)^{\frac{1}{2}} \) | 4. | \(\left(\frac{G M}{R}\right)^{\frac{1}{2}}\) |
The earth is assumed to be a sphere of radius \(R\). A platform is arranged at a height \(R\) from the surface of the earth. The escape velocity of a body from this platform is \(fv_e\), where \(v_e\) is its escape velocity from the surface of the earth. The value of \(f\) is:
1. \(\sqrt{2}\)
2. \(\frac{1}{\sqrt{2}}\)
3. \(\frac{1}{3}\)
4. \(\frac{1}{2}\)
Two satellites of Earth, \(S_1\), and \(S_2\), are moving in the same orbit. The mass of \(S_1\) is four times the mass of \(S_2\). Which one of the following statements is true?
1. | The time period of \(S_1\) is four times that of \(S_2\). |
2. | The potential energies of the earth and satellite in the two cases are equal. |
3. | \(S_1\) and \(S_2\) are moving at the same speed. |
4. | The kinetic energies of the two satellites are equal. |
The figure shows the elliptical orbit of a planet \(m\) about the sun \(\mathrm{S}.\) The shaded area \(\mathrm{SCD}\) is twice the shaded area \(\mathrm{SAB}.\) If \(t_1\) is the time for the planet to move from \(\mathrm{C}\) to \(\mathrm{D}\) and \(t_2\) is the time to move from \(\mathrm{A}\) to \(\mathrm{B},\) then:
1. | \(t_1>t_2\) | 2. | \(t_1=4t_2\) |
3. | \(t_1=2t_2\) | 4. | \(t_1=t_2\) |