Q. No.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:
1. \( \sqrt{2 \sqrt{2} \frac{{GM}}{{R}}} \)
2. \( \sqrt{\frac{{GM}}{{R}}(1+2 \sqrt{2})} \)
3. \( \frac{1}{2} \sqrt{\frac{{GM}}{{R}}(1+2 \sqrt{2})} \)
4. \( \sqrt{\frac{{GM}}{{R}}}\)
1. \({T}\propto{r}\)
2. \({T}\propto\sqrt{{r}}\)
3. \({T}\propto{r}^2\)
4. \({T}^2 \propto{r}^3\)
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A satellite is revolving in a circular orbit at a height \(h\) from the earth's surface (radius of earth \(R\); \(h<<R\)). 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 the atmosphere.)
1. \(\sqrt{2gR}\)
2. \(\sqrt{gR}\)
3. \(\sqrt{\frac{gR}{2}}\)
4. \(\sqrt{gR}\left(\sqrt{2}-1\right)\)
(The radius of the Earth is \(6400~\text{km}\) and \({g}=10~\text{m/s}^2)\)
1. \(0.28\times10^{-3}~\text{rad/s}\)
2. \(1.1\times10^{-3}~\text{rad/s}\)
3. \(0.83\times10^{-3}~\text{rad/s}\)
4. \(0.63\times10^{-3}~\text{rad/s}\)
1. \(+\frac{Gm}{6R}\)
2. \(-\frac{GMm}{2R}\)
3. \(-\frac{GMm}{6R}\)
4. \(\frac{GMm}{2R}\)
1. \(6 \)
2. \(10^{-2} \)
3. \(2 \)
4. \(0.6\)
1. \(2 × 10^{–2}\)
2. \(6 × 10^{–2}\)
3. \(3 × 10^{–2}\)
4. \(10^{–2}\)
Four identical particles of mass \(M\) are located at the corners of a square of side ‘\(a\)’. What should be their speed if each of them revolves under the influence of another gravitational field in a circular orbit circumscribing the square?
| 1. | \(1. 16 \sqrt{\dfrac{G M}{a}} \) | 2. | \(1. 21 \sqrt{\dfrac{G M}{a}} \) |
| 3. | \(1.35 \sqrt{\dfrac{G M}{a}} \) | 4. | \(1. 41 \sqrt{\dfrac{G M}{a}}\) |
A rocket has to be launched from earth in such a way that it never returns. If \(E\) is the minimum energy delivered by the rocket launcher, what should be the minimum energy that the launcher should have if the same rocket is to be launched from the surface of the moon? Assume that the density of the earth and the moon are equal and that of earth's volume is \(64\) times the volume of the moon.
1. \( \frac{E}{4} \)
2. \(\frac{E}{32} \)
3. \(\frac{E}{16} \)
4. \(\frac{E}{64}\)
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