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Q. No.Kepler's third law states that the square of the period of revolution (\(T\)) of a planet around the sun, is proportional to the third power of average distance \(r\) between the sun and planet i.e. \(T^2 = Kr^3\), here \(K\) is constant. If the masses of the sun and planet are \(M\) and \(m\) respectively, then as per Newton's law of gravitation, the force of attraction between them is \(F = \frac{GMm}{r^2},\) here \(G\) is the gravitational constant. The relation between \(G\) and \(K\) is described as:
1. \(GK = 4\pi^2\)
2. \(GMK = 4\pi^2\)
3. \(K =G\)
4. \(K = \frac{1}{G}\)
The kinetic energies of a planet in an elliptical orbit around the Sun, at positions \(A,B~\text{and}~C\) are \(K_A, K_B~\text{and}~K_C\) respectively. \(AC\) is the major axis and \(SB\) is perpendicular to \(AC\) at the position of the Sun \(S\), as shown in the figure. Then:
1. \(K_A <K_B< K_C\)
2. \(K_A >K_B> K_C\)
3. \(K_B <K_A< K_C\)
4. \(K_B >K_A> K_C\)
If \(A\) is the areal velocity of a planet of mass \(M,\) then its angular momentum is:
| 1. | \(\frac{M}{A}\) | 2. | \(2MA\) |
| 3. | \(A^2M\) | 4. | \(AM^2\) |
A planet moves around the sun. At a point \(P,\) it is closest to the sun at a distance \(d_1\) and has speed \(v_1.\) At another point \(Q,\) when it is farthest from the sun at distance \(d_2,\) its speed will be:
| 1. | \(\dfrac{d_2v_1}{d_1}\) | 2. | \(\dfrac{d_1v_1}{d_2}\) |
| 3. | \(\dfrac{d_1^2v_1}{d_2}\) | 4. | \(\dfrac{d_2^2v_1}{d_1}\) |
\(T\) is the time period of revolution of a planet revolving around the sun in an orbit of mean radius \(R\). Identify the incorrect graph.
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2. |  |
| 3. |  |
4. |  |
A planet revolving in elliptical orbit has:
| (A) | a constant velocity of revolution. |
| (B) | the least velocity when it is nearest to the sun. |
| (C) | its areal velocity directly proportional to its velocity. |
| (D) | its areal velocity inversely proportional to its velocity. |
| (E) | to follow a trajectory such that the areal velocity is constant. |
Choose the correct answer from the options given below:
| 1. | (A) only | 2. | (D) only |
| 3. | (C) only | 4. | (E) only |
1. \(2 r_{A}^2=r_{B}^2 \)
2. \(r_{A}^3=2 r_{B}^3 \)
3. \(r_{A}^3=4{r}_{B}^3 \)
4. \(T_{A}^2-{T}_{B}^2=\dfrac{\pi^2}{GM}\left({r}_{B}^3-4 {r}_{A}^3\right)\)
| Statement I: | The kinetic energy of a planet is maximum when it is closest to the sun. |
| Statement II: | The time taken by a planet to move from the closest position (perihelion) to the farthest position (aphelion) is larger for a planet that is farther from the sun. |
| 1. | Statement I is incorrect and Statement II is correct. |
| 2. | Both Statement I and Statement II are correct. |
| 3. | Both Statement I and Statement II are incorrect. |
| 4. | Statement I is correct and Statement II is incorrect. |
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