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Physics · GRAVITATION
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Physics · GRAVITATION
If the semi-major axis of the orbit of a planet is doubled, by what factor does its orbital time period of revolution around the Sun increase?
Read answer →Physics · GRAVITATION
Consider the following statements regarding planetary motion around the Sun: (A) The orbit of a planet is an ellipse with the Sun situated at one of the foci. (B) The areal velocity \( \frac{\Delta A}{\Delta t} \) of a planet is directly proportional to its mass \( m \). (C) The square of the orbital period is proportional to the cube of the semi-major axis. (D) The linear speed of a planet remains constant at all points on its elliptical orbit. Which of the above statements are correct?
Read answer →Physics · GRAVITATION
At what depth \( d \) below the Earth's surface does the acceleration due to gravity reduce to half of its value at the Earth's surface \( g \)? (Take radius of Earth as \( R_E \))
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Three equal masses of mass \( m \) each are situated at the vertices of an equilateral triangle \( ABC \) of side length such that the distance from each vertex to the centroid \( G \) is \( 1\text{ m} \). If the mass at vertex \( A \) is doubled to \( 2m \), what is the magnitude of the net gravitational force on a mass \( 2m \) placed at centroid \( G \)?
Read answer →Physics · GRAVITATION
The orbital speed of a satellite revolving in a circular orbit close to the surface of the Earth (\( h \ll R_E \)) is given by:
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A satellite moves from a circular orbit of radius \( r_1 = 4 R_E \) to a lower circular orbit of radius \( r_2 = R_E \). The ratio of its final orbital speed to its initial orbital speed is:
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Given below are two statements: Statement I: Kepler's second law (law of areas) states that the line joining a planet to the Sun sweeps out equal areas in equal intervals of time. Statement II: Kepler's second law is a consequence of the conservation of linear momentum of the planet. In the light of the above statements, choose the most appropriate answer from the options given below:
Read answer →Physics · GRAVITATION
The work done in lifting a body from height \( h_1 \) to height \( h_2 \) above the Earth's surface can be approximated by \( mg(h_2 - h_1) \) only when:
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): A planet moves faster when it is closer to the Sun and slower when it is farther away. Reason (R): The gravitational force on the planet is a central force, so its orbital angular momentum about the Sun remains conserved. In the light of the above statements, choose the correct answer from the options given below:
Read answer →Physics · GRAVITATION
A point mass \( m \) is placed inside a thin, uniform hollow spherical shell of total mass \( M \) and radius \( R \) at a distance \( r = \frac{R}{3} \) from the centre. The gravitational force exerted on the point mass by the shell is:
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Consider the following statements regarding acceleration due to gravity \( g \) of the Earth (assumed spherical with radius \( R_E \)): (A) At a height \( h = R_E \) above the surface, \( g(h) = \frac{g}{4} \). (B) At a depth \( d = \frac{R_E}{2} \) below the surface, \( g(d) = \frac{g}{2} \). (C) At the centre of the Earth, the acceleration due to gravity is zero. (D) The value of \( g \) at a depth \( d \) is given by \( g(1 - \frac{2d}{R_E}) \) for all \( d < R_E \). Which of the above statements are correct?
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Three equal masses of mass \( m \) are fixed at the vertices of an equilateral triangle \( ABC \) with distances from centroid \( G \) being \( AG = BG = CG = 1\text{ m} \). If the mass at vertex \( A \) is doubled to \( 2m \), what is the magnitude of the net gravitational force on a mass of \( 2m \) kept at centroid \( G \)?
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At what height \( h \) above the surface of the Earth does the acceleration due to gravity become equal to the acceleration due to gravity at a depth \( d = \frac{R_E}{4} \) below the surface of the Earth? (where \( R_E \) is the radius of the Earth)
Read answer →Physics · GRAVITATION
The quotient \( Q = \frac{T^2}{a^3} \) for planetary orbits in the Solar System is approximately \( 3.0 \times 10^{-34}\text{ y}^2\text{ m}^{-3} \). If a newly discovered dwarf planet has an orbital period of \( 64\text{ years} \), its semi-major axis is approximately:
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Three particles of mass \( m \) each are situated at the vertices \( A, B, C \) of an equilateral triangle, with centroid at \( G \), as shown in the diagram. If a particle of mass \( 2m \) is placed at centroid \( G \), what is the resultant gravitational force on this mass at \( G \)?
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In Cavendish's experiment to measure \( G \), a bar of length \( L \) connects two small lead spheres each of mass \( m \). Two large spheres each of mass \( M \) are placed at a separation \( d \) from the small spheres. If the restoring couple per unit angle of twist of the suspension wire is \( \tau_0 \) and the angle of deflection is \( \theta \), what is the equilibrium relation?
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In a Cavendish experiment, a light bar of length \( L = 1.0\text{ m} \) carries two lead spheres of mass \( m = 1.0\text{ kg} \) each at its ends. Two large spheres of mass \( M = 100\text{ kg} \) each are placed at a separation distance \( d = 0.1\text{ m} \) from each small sphere. If the restoring couple per unit angle of the suspension wire is \( \tau = 6.67 \times 10^{-7}\text{ N}\cdot\text{m/rad} \), what is the equilibrium angle of twist \( \theta \)?
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A planet moves in an elliptical orbit around the Sun. If the speed of the planet at perihelion (distance \( r_P \)) is \( v_P \), what is its speed \( v_A \) at aphelion (distance \( r_A \))?
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The orbital radius of the Moon is \( R_m \approx 3.84 \times 10^8\text{ m} \) and its time period is \( T \approx 27.3\text{ days} \approx 2.36 \times 10^6\text{ s} \). The ratio of Earth's surface gravity \( g \approx 9.8\text{ m/s}^2 \) to the centripetal acceleration of the Moon \( a_m \) is approximately:
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For a satellite in a circular orbit very close to the surface of the Earth (\( h \ll R_E \)), the ratio of its orbital speed \( V \) to the escape speed \( v_e \) from the surface of the Earth is:
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Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): The Moon has no atmosphere. Reason (R): The escape speed on the Moon's surface is approximately \( 2.3\text{ km/s} \), which is smaller than the thermal velocities of many gas molecules. In the light of the above statements, choose the correct answer from the options given below:
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According to Kepler's third law, the ratio \( T^2 / a^3 \) for planetary motion around the Sun is approximately constant. What is the value of this quotient in units of \( 10^{-34} \text{ y}^2 \text{ m}^{-3} \)?
Read answer →Physics · GRAVITATION
