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Q. No.A body of mass \(m\) is moving in a circular orbit of radius \(R\) about a planet of mass \(M.\) At some instant, it splits into two equal masses. The first mass moves in a circular orbit of radius \(\frac{R}{2},\) and the other mass, in a circular orbit of radius \(\frac{3R}{2}.\) The difference between the final and initial total energies is:
1. \(+\frac{Gm}{6R}\)
2. \(-\frac{GMm}{2R}\)
3. \(-\frac{GMm}{6R}\)
4. \(\frac{GMm}{2R}\)
1. \(+\frac{Gm}{6R}\)
2. \(-\frac{GMm}{2R}\)
3. \(-\frac{GMm}{6R}\)
4. \(\frac{GMm}{2R}\)
Subtopic: Gravitational Potential Energy |
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The energy required to take a satellite to a height \({'h'}\) above the Earth's surface \(\text{(radius of Earth}~ R=6.4 \times 10^3~\mathrm{km})~\text{is} ~{E}_1\) and kinetic energy required for the satellite to be in a circular orbit at this height is \({E}_2.\) The value of \({h}\) for which \({E_1~\text{and}~E_2}\) are equal is:
1. \(1.6 \times 10^3~\text{km}\)
2. \(3.2 \times 10^3~\text{km}\)
3. \(6.4 \times 10^3~\text{km}\)
4. \(1.28 \times 10^4~\text{km}\)
1. \(1.6 \times 10^3~\text{km}\)
2. \(3.2 \times 10^3~\text{km}\)
3. \(6.4 \times 10^3~\text{km}\)
4. \(1.28 \times 10^4~\text{km}\)
Subtopic: Gravitational Potential Energy |
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Two planets have masses \(M\) and \(16M\) and their radii are \(a\) and \(2a\), respectively. The separation between the centres of the planets is \(10a\). A body of mass \(m\) is fired from the surface of the larger planet towards the smaller planet along the line joining their centres. For the body to be able to reach at the surface of smaller planet, the minimum firing speed needed is:
1. \( \sqrt{\frac{G M^2}{m a}} \)
2. \( \frac{3}{2} \sqrt{\frac{5 G M}{a}} \)
3. \( 4 \sqrt{\frac{G M}{ma}} \)
4. \(4 \sqrt{\frac{G M}{a}} \)
Subtopic: Gravitational Potential Energy |
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Four spheres each of mass \(m\) form a square of side \(d\) (as shown in the figure). A fifth sphere of mass \(M\) is situated at the centre of the square. The total gravitational potential energy of the system is:
1. \( -\frac{{Gm}}{{d}}[(4+\sqrt{2}) {m}+4 \sqrt{2} {M}] \)
2. \(-\frac{{Gm}}{{d}}[(4+\sqrt{2}) {M}+4 \sqrt{2}{m}] \)
3. \( -\frac{{Gm}}{{d}}\left[3 {m}^2+4 \sqrt{2}{M}\right] \)
4. \(-\frac{{Gm}}{{d}}\left[6{m}^2+4 \sqrt{2}{M}\right]\)
1. \( -\frac{{Gm}}{{d}}[(4+\sqrt{2}) {m}+4 \sqrt{2} {M}] \)
2. \(-\frac{{Gm}}{{d}}[(4+\sqrt{2}) {M}+4 \sqrt{2}{m}] \)
3. \( -\frac{{Gm}}{{d}}\left[3 {m}^2+4 \sqrt{2}{M}\right] \)
4. \(-\frac{{Gm}}{{d}}\left[6{m}^2+4 \sqrt{2}{M}\right]\)
Subtopic: Gravitational Potential Energy |
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A body is projected vertically upwards from the surface of the earth with a velocity equal to one-third of escape velocity. The maximum height attained by the body will be:
(take the radius of the Earth \(R=6400~\text{km}\) and \(g=10~\text{m/s}^2\) )
1. \(800~\text{km}\)
2. \(1600~\text{km}\)
3. \(2133~\text{km}\)
4. \(4800~\text{km}\)
(take the radius of the Earth \(R=6400~\text{km}\) and \(g=10~\text{m/s}^2\) )
1. \(800~\text{km}\)
2. \(1600~\text{km}\)
3. \(2133~\text{km}\)
4. \(4800~\text{km}\)
Subtopic: Gravitational Potential Energy |
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An object of mass \(1\) kg is taken to a height from the surface of the Earth which is equal to three times the Earth's radius. The gain in potential energy of the object will be:
(Given: \(g=10\) ms–2 and radius of Earth \(=6400\) km)
1. \(48\) MJ
2. \(24\) MJ
3. \(36\) MJ
4. \(12\) MJ
(Given: \(g=10\) ms–2 and radius of Earth \(=6400\) km)
1. \(48\) MJ
2. \(24\) MJ
3. \(36\) MJ
4. \(12\) MJ
Subtopic: Gravitational Potential Energy |
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An object of mass \(m\) is placed at a height \(R_{e}\) from the surface of the earth. What is the increase in potential energy of the object if the height of the object is increased to \(2R_{e}\) from the surface? (\(R_{e}:\) Radius of the earth)
1. \({\dfrac{1}{3}{mgR}_{e}}\)
2. \({\dfrac{1}{6}{mgR}_{e}}\)
3. \({\dfrac{1}{2}{mgR}_{e}}\)
4. \({\dfrac{1}{4}{mgR}_{e}}\)
1. \({\dfrac{1}{3}{mgR}_{e}}\)
2. \({\dfrac{1}{6}{mgR}_{e}}\)
3. \({\dfrac{1}{2}{mgR}_{e}}\)
4. \({\dfrac{1}{4}{mgR}_{e}}\)
Subtopic: Gravitational Potential Energy |
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A particle is released at a height equal to the radius of the earth above the surface of the earth. Its velocity when it hits the surface of the earth is equal to:
(Where, \(M_e\)=mass of earth, \(R_e\)= radius of the earth.)
1. \(\sqrt{\left[{\frac{2{GM}_{e}}{{R}_{e}}}\right]}\)
2. \(\sqrt{\left[{\frac{{GM}_{e}}{2{R}_{e}}}\right]}\)
3. \(\sqrt{\left[{\frac{{GM}_{e}}{{R}_{e}}}\right]}\)
4. \(\sqrt{\left[{\frac{2{GM}_{e}}{3{R}_{e}}}\right]}\)
(Where, \(M_e\)=mass of earth, \(R_e\)= radius of the earth.)
1. \(\sqrt{\left[{\frac{2{GM}_{e}}{{R}_{e}}}\right]}\)
2. \(\sqrt{\left[{\frac{{GM}_{e}}{2{R}_{e}}}\right]}\)
3. \(\sqrt{\left[{\frac{{GM}_{e}}{{R}_{e}}}\right]}\)
4. \(\sqrt{\left[{\frac{2{GM}_{e}}{3{R}_{e}}}\right]}\)
Subtopic: Gravitational Potential Energy |
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A particle is released from a height equal to radius of earth, \(R.\) Its velocity when it strikes the ground is:
1. \(\sqrt{gR}\)
2. \(\sqrt{{{gR}\over{2}}}\)
3. \(\sqrt{2gR}\)
4. \(\sqrt{4gR}\)
1. \(\sqrt{gR}\)
2. \(\sqrt{{{gR}\over{2}}}\)
3. \(\sqrt{2gR}\)
4. \(\sqrt{4gR}\)
Subtopic: Gravitational Potential Energy |
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