Q. No.A planet moves around the sun in an elliptical orbit with the perihelion at \(P ,\) and aphelion at \(A\). Let the quantities be defined as follows: (for the planet)
\(r\)
distance from sun \(S\)
\(v\)
speed in orbit
\(K\)
kinetic energy
\(U\)
potential energy
The subscripts refer to the quantity measured at the perihelion \((P)\) or aphelion \((A)\): \(v_P\) is the speed at perihelion, \(K_A\) is the kinetic energy at aphelion, etc. Then,
1. \(K_A r^2_A = K_Pr^2_P\)
2. \(v_A r_A = v_P~r_P\)
3. \(U_Ar_A = U_P r_P\)
4. All the above are true
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| 1. | \(2\) km/s | 2. | \(2\sqrt2\) km/s |
| 3. | \(2(\sqrt2-1)\) km/s | 4. | \(2(\sqrt2+1)\) km/s |
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| 1. | \(4\) | 2. | \(2\) |
| 3. | \(\dfrac12\) | 4. | \(\dfrac14\) |
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| Statement I: | The gravitational force exerted by the Sun on the Earth is reduced when the Moon is between the Earth and the Sun. |
| Statement II: | The gravitational force exerted by the Sun on the Earth is reduced when the Moon is opposite to the Sun, relative to the Earth. |
| 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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| Statement I: | The gravitational force acting on a particle depends on the electric charge of the particle. |
| Statement II: | The gravitational force on an extended body can be calculated by assuming the body to be a particle 'concentrated' at its centre of mass and applying Newton's law of gravitation. |
| 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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| 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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| 1. | \(3F\) | 2. | \(2F\) |
| 3. | \(4F\) | 4. | \(F\) |
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| 1. | \(\dfrac{GMm}{r^2}+\dfrac{2Gm^2}{r^2}\) |
| 2. | \(\dfrac{GMm}{r^2}+\dfrac{\sqrt3Gm^2}{r^2}\) |
| 3. | \(\dfrac{GMm}{r^2}+\dfrac{Gm^2}{\sqrt3r^2}\) |
| 4. | \(\dfrac{GMm}{r^2}+\dfrac{2Gm^2}{\sqrt3r^2}\) |
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| 1. | \(mr\) | 2. | \(\dfrac{m}{r}\) |
| 3. | \(m\sqrt r\) | 4. | \(\dfrac{m}{\sqrt r}\) |
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| 1. | \(U > mgh\) |
| 2. | \(U < mgh\) |
| 3. | \(U = mgh\) |
| 4. | any of the above may be true depending on the value of \(h,\) considered relative to the radius of the earth. |
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