Newton's First Law tells us that an object in motion at a constant speed in a straight line will continue to move in a straight line at that constant speed unless the object experiences an unbalanced external force. In the case of the motion of a rock tied to a string and swung in a circular path at a constant speed (commonly referred to as uniform circular motion) one of these conditions is met in that the speed is held constant. However, the motion is not in a straight line, so you may wonder about the nature of the force that is required by Newton's First Law. What supplies that force? How can we describe it? Can it be calculated from the information we have about the motion? The direction in which a rock on the end of a string moves is obviously changing as the rock moves in its circular path. Perhaps that can provide a clue as to the direction of the required force.

A simple experiment will demonstrate the source of the force. If the string is cut or if you release the string, the rock no longer moves in a circular path, clearly indicating that the string provides the force necessary to change the direction of motion. If you observe carefully you may note that the rock moves in a direction tangent to the circle when the string is released. Analysis of the velocity vectors for several positions of the rock during the motion leads to the conclusion that the force and acceleration are directed toward the center of the circle, hence the names centripetal force and centripetal acceleration. Additional experiments with different radii and different speeds show that the acceleration increases with the square of the speed and is inversely proportional to the radius of the circular path. The expression for this centripetal acceleration is a_c = v² / r where v is the speed and r is the radius of the circle.

Anytime there is motion in a circular path at a constant speed, the force necessary to provide the motion can be calculated as the product of the object's mass and the centripetal acceleration. In some cases the force is provided by direct contact, such as that of the string attached to the rock, while in other cases there is no need for direct contact, such as is the case with the gravitational force on a satellite.

Planets do not move in perfect circles about the Sun; they move in an elliptical path that looks like a circle that is elongated in one direction. Kepler formulated three laws that describe how the planets rotate about the Sun. The first law tells us that the Sun is located at the focus of the elliptical path. This law determines the orbital path of the planet. The second law tells us that a line drawn between the focus and the planet sweeps out equal areas in equal time. As a consequence of this law a planet moves faster when it is closer to the Sun and slower when it is further away. In Chapter 8 you will learn that Kepler's second law is connected with a fundamental relationship in physics known as conservation of angular momentum. The third law tells us that the cube of the radius about the Sun for each planet is proportional to the square of the period of the orbit. This law enables us to calculate the distance of a planet from the Sun if we know the period or vice versa.

Early models of the motion of the planets, notably that developed by Ptolemy, treated the Earth as the center of planetary motion, in part because of the belief that the Earth was the center of creation. As more data were obtained regarding the motion of the planets, it became more difficult to use such a geocentric or "Earth-centered" model to predict the motion of the planets. Copernicus developed a heliocentric or "Sun-centric" model in which the planets, including the Earth, were considered to be rotating about the Sun. This view was considered to be quite radical at the time when Copernicus proposed it due, in part, to the fact that the religious leaders of the time believed in the geocentric model. The model that was proposed by Copernicus considered the orbits of the planets about the Sun to be circles, and it was Kepler who refined the model into its present day form by proposing elliptical orbits for the planets.

In Chapter 4 you studied the three laws attributed to Isaac Newton that describe motion. In this chapter you learn that Newton is also credited with formulating the fundamental law of gravity. Newton's Law of Universal Gravitation applies to all objects in the universe. It describes the mutual attraction between objects as depending upon the product of their masses. It also states that the force of attraction decreases with the square of the separation distance between the centers of the objects. Such a dependence is written as 1 / r² where r is the separation distance between the centers of the objects and is referred to as an inverse square relationship. We will encounter inverse square relationships in other areas of physics in subsequent chapters of this text. Written in equation form by including the constant that makes the proportionality an equality, Newton's Law of Universal Gravitation is F = G m₁ m₂ / r². It is this attraction of the Earth for objects that was expressed as the acceleration due to gravity, g = 9.8 m / s², in earlier chapters. Thus your weight represents the attraction you experience toward the center of the Earth. You exert an equal and oppositely directed force of attraction on the Earth.

The centripetal force necessary to keep planets in motion about the Sun and to keep the Moon in motion about the Earth, and a man-made satellite in motion about the Earth is provided by the gravitational force of attraction between the two respective objects. The gravitational attraction of the Sun for the Earth keeps the Earth in an orbit about the Sun just as the string on the rock keeps the rock in orbit around your hand. Gravity is a force that can produce "action at a distance" without having any direct contact between the objects such as the string between your hand and the rotating rock. We will encounter other "action at a distance" forces when we study electricity and magnetism in Chapters 12 and 14.