Gravitation
Class 11 Physics · Physics Part I · Chapter 7
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Summary
This chapter develops the universal law of gravitation from Kepler's laws of planetary motion: planets move in elliptical orbits with the Sun at a focus (law of orbits); the line joining a planet to the Sun sweeps equal areas in equal times (law of areas, a consequence of conservation of angular momentum); and the square of the orbital period is proportional to the cube of the semi-major axis (law of periods). Newton's universal law states that every body attracts every other with a force \(F=Gm_1m_2/r^{2}\), directed along the line joining them, where \(G=6.67\times10^{-11}\ \text{N m}^{2}\text{kg}^{-2}\) was measured by Cavendish. Acceleration due to gravity \(g=GM_E/R_E^{2}\) decreases with height as \(g(1-2h/R_E)\) and with depth as \(g(1-d/R_E)\). Gravitational potential energy of two masses is \(-Gm_1m_2/r\), taken zero at infinity. The escape speed from Earth is \(v_e=\sqrt{2gR_E}\approx11.2\ \text{km s}^{-1}\). Earth satellites obey Kepler's laws; the orbital speed is \(\sqrt{GM_E/(R_E+h)}\) and the total energy of an orbiting satellite is negative, \(-Gm M_E/2(R_E+h)\), confirming a bound orbit. A uniform spherical shell exerts no gravitational force on a mass inside it.
Kepler's lawsUniversal law of gravitationThe gravitational constantAcceleration due to gravity above and below the surfaceGravitational potential energyEscape speedEarth satellites and energy of an orbiting satellite
Key terms
Kepler's law of areas
The line joining a planet to the Sun sweeps out equal areas in equal time intervals, a consequence of angular momentum conservation.
Universal gravitational constant
The constant \(G=6.67\times10^{-11}\ \text{N m}^{2}\text{kg}^{-2}\) in Newton's law of gravitation, first measured by Cavendish.
Acceleration due to gravity
The acceleration of a freely falling body near a planet's surface, \(g=GM/R^{2}\).
Gravitational potential energy
The energy of a system of masses due to their positions, \(U=-Gm_1m_2/r\), taken zero at infinity.
Escape speed
The minimum speed needed to project a body so that it never returns, \(v_e=\sqrt{2gR}\approx11.2\ \text{km s}^{-1}\) for Earth.
Orbital velocity
The speed of a satellite in a circular orbit, \(v=\sqrt{GM/(R+h)}\).
Extra questions & answers
Try one from this chapter
Kepler's law of areas is a consequence of the conservation of: