Q. No.
The acceleration due to gravity, \(g\), near a spherically symmetric planet's surface decreases with height, \(h\) according to the relation:
\(g(h)= g_s-k\cdot h\), where \(h\ll\) the radius of the planet.
The escape speed from the planet's surface is:
1.
\(\dfrac{g_s}{2\sqrt k}\)
2.
\(\dfrac{g_s}{\sqrt k}\)
3.
\(\dfrac{2g_s}{\sqrt k}\)
4.
\(g_s\sqrt{\dfrac{2}{k}} \)
\(g(h)= g_s-k\cdot h\), where \(h\ll\) the radius of the planet.
The escape speed from the planet's surface is:
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If a particle is projected vertically upward with a speed \(u,\) and rises to a maximum altitude \(h\) above the earth's surface then:
(\(g=\) acceleration due to gravity at the surface)
| 1. | \(h>\dfrac{u^2}{2g}\) |
| 2. | \(h=\dfrac{u^2}{2g}\) |
| 3. | \(h<\dfrac{u^2}{2g}\) |
| 4. | Any of the above may be true, depending on the earth's radius |
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Consider a satellite orbiting the Earth in a circular orbit. Then,
| 1. | The gravitational force on the satellite is the centripetal force. |
| 2. | The gravitational force on the satellite is the centrifugal force. |
| 3. | The gravitational force on the satellite is greater than the centripetal force. |
| 4. | All the above are true |
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| 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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