Q. No.Two particles of masses \(m_1,~m_2\) are placed on the axis of a uniform circular ring of mass \(M\) and radius \(R,\) on opposite sides of the centre of the ring. The distances of \(m_1,~m_2\) from the centre of the ring are \(x_1,~x_2\) respectively, and \(x_1~ x_2 \ll R.\) The net force on the ring vanishes. Then,
1.
\(\dfrac{m_{1}}{x_{1}}=\dfrac{m_{2}}{x_{2}} \)
2.
\(\dfrac{m_{1}}{x_{1}^{2}}=\dfrac{m_{2}}{x_{2}^{2}} \)
3.
\(\dfrac{m_{1}}{x_{1}^{3}}=\dfrac{m_{2}}{x_{2}^{3}} \)
4.
\(m_{1} x_{1}=m_{2} x_{2} \)
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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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| 1. | \(mr\) | 2. | \(\dfrac{m}{r}\) |
| 3. | \(m\sqrt r\) | 4. | \(\dfrac{m}{\sqrt r}\) |
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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. | \(3F\) | 2. | \(2F\) |
| 3. | \(4F\) | 4. | \(F\) |
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| \(r\) | 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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