Imagine that we are trying to calculate the gravitational force between the Earth and its satellites, we have to be careful about the distance r used. This is true for a system of planets where the distance between a planet (say, the Earth) to the Sun is sufficiently large compared to the sizes of the celestial bodies. One of the assumptions is that the two masses are both point masses that is, their geometrical size does not matter. However, in deriving this formula, some assumptions were made. Here, G is the universal gravitational constant G = 6.674 × 10 − 11 undefined N m 2 /kg 2. In what way would the force of gravity have to depend on mass for the acceleration of a body to be proportional to its weight, as Aristotle contended?į = G m 1 m 2 r 2, which essentially states that the force F between two masses m 1 and m 2 is inversely proportional to the square of the distance r. Compare your conclusion with Galileo's conclusion that all freely falling objects on the surface of the earth undergo the same acceleration. How would the motion of bodies at the surface of the earth differ if the force of gravity increased with altitude? 16.ĭiscuss the motion of freely falling bodies on the surface of a fictitious planet whose force of gravity is independent of the mass of the falling body. Increases in altitude are accompanied by decreases in the force of gravity. Compare (a) their centripetal accelerations, and (b) the forces of gravity exerted on each by the earth. Two satellites whose masses are m 1 and m 2 circle the earth in an orbit of radius R. The force of gravity is proportional to the mass of a body, yet all freely falling bodies near the surface of the earth experience the same acceleration. To provide insight we now introduce the universal law of gravity by exhibiting its relationships to the results of Kepler and Galileo. He built this law on the foundation provided by Kepler's laws of planetary motion and Galileo's studies of terrestrial motion.
Newton filled this gap by formulating the universal law of gravity. In this sense Newton's second law is incomplete. However, Newton's second law does not describe any specific force in terms of the properties of the particle and of its environment. The acceleration of the moon is “explained” in a similar way. Thus a falling apple accelerates as a result of the force exerted on it by the earth. In other words, the same laws of motion-Newton's laws of motion-govern both celestial and terrestrial motion.Īccording to Newton's second law, the net force acting on a particle “causes” it to accelerate. Newton recognized that the laws of motion must also be universal in character. Newton's realization that gravity is universal had important implications. The apple is then in orbit! Newton recognized that the same force that acts on the apple-gravity-also acts on the moon. For example, as the horizontal velocity of a falling apple increases, it reaches a point where, because of the curvature of the earth, the earth recedes from the apple at the same rate that the apple falls toward the earth.
Newton saw the orbiting moon as the limiting case of a falling object possessing a large horizontal velocity. In Newton's hands, however, universal gravitation and a universal set of dynamic laws evolved together. The motion of falling apples, for example, was never connected to the motion of an orbiting planet. Prior to Newton, scientists believed that physics on the earth differed from physics in the heavens. This, too, remained for Newton to discover. However, Kepler failed in his attempt to find this long-range guiding force. Kepler had studied the planets and discovered three laws describing their motion. The unknown force guiding planets in their orbits acted over great distances. However, the explanation of these gravitational effects remained for Newton to discover.Īn apparently unrelated problem of 17th-century science involved celestial motion. Galileo had already formulated the laws of kinematics, which described the motion of objects influenced by gravity. Gravity was thought to be a local force, and only those objects near the earth were believed to be affected by it. In Newton's day the word gravity referred to the force exerted on objects by the earth.
Joseph Priest, in International Edition University Physics, 1984 6.6 The Universal Force of Gravitation