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Q. No.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 |
Subtopic: Satellite |
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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. | \(g_s\sqrt{\dfrac{2}{k}} \) |
Subtopic: Escape velocity |
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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 |
Subtopic: Acceleration due to Gravity |
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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. | \(m_{1} x_{1}=m_{2} x_{2} \) |
Subtopic: Newton's Law of Gravitation |
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The gravitational potential energy of a particle of mass \(m\) increases by \(mgh,\) when it is raised through a height \(h\) in a uniform gravitational field "\(g\)". If a particle of mass \(m\) is raised through a height \(h\) in the earth's gravitational field (\(g\): the field on the earth's surface) and the increase in gravitational potential energy is \(U\), then:
| 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. |
Subtopic: Gravitational Potential Energy |
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The angular momentum of a planet of mass \(m,\) moving around the sun (mass: \(M\gg m\)) in an orbit of radius \(r\) is proportional to:
| 1. | \(mr\) | 2. | \(\dfrac{m}{r}\) |
| 3. | \(m\sqrt r\) | 4. | \(\dfrac{m}{\sqrt r}\) |
Subtopic: Kepler's Laws |
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Three stars of identical masses \(m\) move around a central star of mass \(M\) in an orbit of radius \(r.\) The net gravitational force acting on any one of the orbiting stars equals:
| 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. | \(\dfrac{GMm}{r^2}+\dfrac{2Gm^2}{\sqrt3r^2}\) |
Subtopic: Newton's Law of Gravitation |
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A particle of mass \(m\) is placed at the mid-point of the radius of a thin uniform spherical shell of mass \(M,\) as shown in the figure. Consider the plane that slices the shell into two parts: the plane is perpendicular to the radius and passes through \(m.\) The upper part of the shell has a mass \(\dfrac M4\) and the lower part \(\dfrac{3M}{4}.\) Let the gravitational force exerted by the upper part of the shell on the particle be \(F.\) The force exerted by the lower part of the shell on the particle is:
| 1. | \(3F\) | 2. | \(2F\) |
| 3. | \(4F\) | 4. | \(F\) |
Subtopic: Newton's Law of Gravitation |
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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)
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
| \(r\) | 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
Subtopic: Kepler's Laws |
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A satellite is in a circular orbit around a planet, orbiting with a speed of \(2\) km/s. What is the minimum additional velocity that should be given to it, perpendicular to its motion, so that it escapes?
| 1. | \(2\) km/s | 2. | \(2\sqrt2\) km/s |
| 3. | \(2(\sqrt2-1)\) km/s | 4. | \(2(\sqrt2+1)\) km/s |
Subtopic: Escape velocity |
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