14 The Law of Gravity

C h a p t e r O u t l i n e 14.1 Newton’s Law of Universal Gravitation 14.2 Measuring the Gravitational Constant 14.3 Free-Fall Acceleration and the Gravitational Force 14.4 Kepler’s Laws 14.5 The Law of Gravity and the Motion of Planets 14.6 The Gravitational Field 14.7 Gravitational Potential Energy 14.8 Energy Considerations in Planetary and Satellite Motion 14.9 (Optional) The Gravitational Force Between an Extended Object and a Particle 14.10 (Optional) The Gravitational Force Between a Particle and a Spherical Mass

14.1 Newton’s Law of Universal Gravitation every particle in the Universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

Properties of the gravitational force Because the force varies as the inverse square of the distance between the particles, it decreases rapidly with increasing separation. the gravitational force exerted by a finite-size, spherically symmetric mass distribution on a particle outside the distribution is the same as if the entire mass of the distribution were concentrated at the center.

14.2 Measuring the Gravitational Constant The universal gravitational constant G was measured in an important experiment by Henry Cavendish (1731–1810) in 1798.

EXAMPLE : Billiards, Anyone?

Solution

14.3 Free-Fall Acceleration and the Gravitational Force

14.4 Kepler’s Laws 1. All planets move in elliptical orbits with the Sun at one focal point. 2. The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals. 3. The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit.

14.5 The Law of Gravity and the Motion of Planets

Kepler’s Third Law

Kepler’s Second Law and Conservation of Angular Momentum the radius vector from the Sun to a planet sweeps out equal areas in equal time intervals.

14.6 The Gravitational Field

14.7 Gravitational Potential Energy the gravitational force is conservative. a central force is always directed along one of the radial segments, therefore, the work done by F along any radial segment is any central force is conservative

Three interacting particles

14.8 Energy Considerations in Planetary and Satellite Motion

The total mechanical energy is negative in the case of circular orbits. The kinetic energy is positive and equal to one-half the absolute value of the potential energy. both the total energy and the total angular momentum of a gravitationally bound, two-body system are constants of the motion.

Escape Speed

14.9 (Optional) The Gravitational Force Between an Extended Object and a Particle

14.10 The Gravitational Force Between a Particle and a Spherical Mass