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Physics · GRAVITATION
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Physics · GRAVITATION
The work done in lifting a particle of mass \( m \) radially from a distance \( r_1 = R_E \) to \( r_2 = 3 R_E \) from the center of the Earth of mass \( M_E \) and radius \( R_E \) is:
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For a satellite of mass \( m \) moving in a circular orbit of radius \( r \) around the Earth, the ratio of its kinetic energy (\( K.E. \)) to its total mechanical energy (\( E \)) is:
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Match List-I with List-II: List-I (Physical Quantity) List-II (Expression / Value) A Escape speed from Earth I \( \approx 85\text{ minutes} \) B Escape speed from Moon II \( \approx 11.2\text{ km s}^{-1} \) C Time period of near-Earth orbit (\( h \ll R_E \)) III \( \approx 2.3\text{ km s}^{-1} \) D Orbital speed of near-Earth satellite IV \( \sqrt{g R_E} \) Choose the correct answer from the options given below:
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A planet of mass \( m \) revolves around the Sun in an elliptical orbit. If the magnitude of its angular momentum about the Sun is \( L \), the areal velocity \( \frac{\Delta A}{\Delta t} \) swept out by the position vector of the planet is given by:
Read answer →Physics · GRAVITATION
Given below are statements regarding planetary motion: (A) Kepler's second law (law of areas) is a direct consequence of the conservation of angular momentum. (B) The linear speed of a planet is maximum when it is at perihelion. (C) A central force exerts zero torque about the centre of force. (D) For any planet, the ratio \( T^2 / a^3 \) depends strongly on the planet's mass \( m \). Choose the correct answer from the options given below:
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A body of mass \( m \) is taken from a depth \( d = \frac{R_E}{2} \) inside the Earth to a height \( h = R_E \) above the surface of the Earth. What is the ratio of the acceleration due to gravity at this depth to that at this height?
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What is the work required to be done by an external agent to slowly move a body of mass \( m \) from the surface of Earth (radius \( R_E \), mass \( M_E \)) to a height \( h = R_E \) above the surface?
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Match List-I with List-II: List-I List-II A Kepler's First Law I \( \frac{\Delta A}{\Delta t} = \text{constant} \) B Kepler's Second Law II \( T^2 \propto a^3 \) C Kepler's Third Law III Measurement of \( G \) D Cavendish Experiment IV Elliptical orbits with Sun at one focus Choose the correct answer from the options given below:
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Given below are two statements: Statement I: The gravitational force between two point masses satisfies Newton's third law of motion in vector form as \( \mathbf{F}_{12} = -\mathbf{F}_{21} \). Statement II: The universal gravitational constant \( G \) has the dimensional formula \( [\text{M}^{-1} \text{L}^3 \text{T}^{-2}] \) and its SI unit is \( \text{N m}^2 \text{kg}^{-2} \). In the light of the above statements, choose the most appropriate answer from the options given below:
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Four point masses, each of mass \( m \), are fixed at the corners of a square of side \( l \). What is the minimum work required to slowly move an additional test mass \( m_0 \) from infinity to the center of the square?
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A satellite revolves around the Earth in an orbit very close to its surface. Taking \( g = 9.8\text{ m/s}^2 \) and \( R_E = 6.4 \times 10^6\text{ m} \), its time period of revolution is approximately:
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Given below are four statements regarding the escape speed from a planet: (A) The escape speed depends on the mass of the projectile. (B) The escape speed is independent of the angle of projection from the planet's surface. (C) Gas molecules on the Moon escape because their thermal velocities can exceed the escape velocity of the Moon. (D) Escape speed from the surface of Earth is \( \sqrt{2} \) times the orbital speed of a satellite orbiting close to Earth. Choose the correct answer from the options given below:
Read answer →Physics · GRAVITATION
According to Kepler's second law, the areal velocity of a planet revolving around the Sun is constant. This is a direct consequence of the conservation of which physical quantity?
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Two uniform solid spheres of radii \( R \) have masses \( M \) and \( 4M \), and their centres are separated by a fixed distance of \( 6R \). The distance from the centre of the sphere of mass \( M \) along the line joining their centres where the net gravitational force on a test particle is zero is:
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Assuming the orbits of Earth and Mars around the Sun are circular, if the orbital radius of Mars is \( 1.52 \) times that of the Earth, and the Earth year is \( 365 \) days, the length of the Martian year is approximately:
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Consider the following statements regarding an orbiting satellite around the Earth: (A) The total mechanical energy of a satellite orbiting at finite distance is always negative. (B) The kinetic energy of a satellite in a circular orbit is equal to half the magnitude of its gravitational potential energy. (C) If the total energy of a satellite becomes zero or positive, it escapes to infinity. (D) In an elliptical orbit, both kinetic energy and potential energy remain constant throughout the orbit. Which of the above statements are correct?
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Consider the following statements regarding the acceleration due to gravity of Earth: (A) At the centre of the Earth, the acceleration due to gravity is zero. (B) For a height \( h \ll R_E \), the acceleration due to gravity is given by \( g(h) \approx g(1 - 2h/R_E) \). (C) The acceleration due to gravity increases linearly as one goes deeper into the Earth. (D) The value of \( g \) is maximum at the surface of the Earth. Which of the above statements are correct?
Read answer →Physics · GRAVITATION
Match List-I with List-II: List-I (Physical Quantity/State) List-II (Formula / Value) A \( g \) at height \( h \ll R_E \) I \( g (1 - \frac{d}{R_E}) \) B \( g \) at depth \( d \) II \( g (1 - \frac{2h}{R_E}) \) C Gravitational force inside uniform spherical shell III \( -G M_E m (\frac{1}{r_2} - \frac{1}{r_1}) \) D Work done \( W_{12} \) in moving mass \( m \) from \( r_1 \) to \( r_2 \) IV Zero Choose the correct answer from the options given below:
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If the acceleration due to gravity at a small height \( h \) (\( h \ll R_E \)) above the Earth's surface equals the acceleration due to gravity at a depth \( d \) below the Earth's surface, the relation between \( d \) and \( h \) is:
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Three identical point masses, each of mass \( m \), are placed at the vertices of an equilateral triangle of side length \( 1\text{ m} \). A point mass \( 2m \) is placed at the centroid of the triangle. What is the net gravitational force experienced by the mass at the centroid?
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If the mean density of the Earth is \( \rho \), the time period of a satellite revolving in an orbit very close to the Earth's surface depends on \( \rho \) as:
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Match List-I with List-II for a satellite of mass \( m \) in a circular orbit of radius \( r = R_E + h \) around Earth of mass \( M_E \): List-I (Energy Parameter) List-II (Formula) A Kinetic Energy I \( \frac{G M_E m}{2r} \) B Potential Energy II \( -\frac{G M_E m}{r} \) C Total Mechanical Energy III \( -\frac{G M_E m}{2r} \) D Binding Energy IV \( \frac{G M_E m}{2r} \) Choose the correct answer from the options given below:
Read answer →Physics · GRAVITATION
