Chapter 5: Circular Motion; GravitationKinematics of Uniform Circular MotionUniform circular motion describes the movement of an object in a circular path at a constant speed, v, but at a constantly changing velocity vector, v, necessary for the object to stay on the path. The velocity vector at a given point in the path is the tangent to the circular path in the direction of motion. The acceleration corresponding to this changing velocity at a given point is directed toward the center of the circle, and it is called centripetal or radial acceleration. Radial acceleration is proportional to the square of the velocity and inversely proportional to the radius, .rvR2 At a point in the path, the velocity and acceleration of an object undergoing uniform circular motion are perpendicular. The number of revolutions per second by the object is denoted by the frequency, f. The time in seconds for each revolution, T, is its reciprocal, , and is called the period.fT1 From these definitions, an equation for speed can be derived, .Trv2Dynamics of Uniform Circular MotionThe force required to keep an object in uniform circular motion is directed inward toward the center and is the product of mass and radial acceleration, .rmvFR2 Were the force to stop being applied , the object would continue in the path of the velocity vector at that point, tangent to the circle of motion. Often the force is provided by the tension of a rope or string. Automobiles require friction between the wheels and the road to provide the inward force necessary for moving in a circle, or more often, in an arc of a circle. Satellites use gravity to provide the centripetal force for circular orbits.Nonuniform Circular MotionWhen the net force on an object is not directed toward the center, the force vector can be broken into two perpendicular components: the inwardly directed radial force, FR, and the tangential force, Ftan. Similarly, the acceleration can be broken into two perpendicular components: its inward radial acceleration, aR, and the tangential acceleration, atan. The change of speed in a nonuniform circular motion is a consequence of the tangential force, and thus of the tangential acceleration. When the speed of circular motion decreases, the velocity vector is antiparallel to the tangential acceleration component. When the speed of circular motion increases, the velocity vector is parallel to the tangential acceleration component. From the Pythagorean theorem, the magnitude of acceleration at a point is equal to the square root of the sum of the squares of the components, .2tan2RNewtons Law of Universal GravitationNewton derived the theory of gravitation with the goal of explaining the motion of the moon around the Earth. He reasoned that the radial force that keeps the Moon moving in an approximately uniform circular motion is related to the force of gravity observed at the Earths surface. Newtons law of universal gravitation states that for both masses, the magnitude of the attractive force between them is proportional to the product of the masses and inversely proportional to the square of distance. This is now represented by , where G is the universal constant, 21rmF, whose value was first approximated by Cavendish a century after Newton 2kgmN1-06.7proposed his law.Gravity Near the Earths Surface; Geophysical ApplicationsNewtons law of universal gravitation is consistent with the equation F=mg at the Earths surface, as g represents the product of the mass of the Earth, 5.981024 kg, and the constant G divided by the square of average radius of Earth, 6.38106 m. Values of g fluctuate slightly across the Earths surface because there is not a single fixd radius and because the distribution of mass in the Earth is not uniform.Satellites and “Weightlessness”Satellites remain in orbit because their tangential speed is sufficient to keep them from falling to the Earths surface, but no so great as to cause them to fly out of their circular or elliptic path. The inward-directed force and acceleration necessary for circular motion is due to gravity. Apparent changes in the force of gravity are influenced by the frame of reference of the observer. That is, if the frame of reference is accelerating in the same direction as gravity, apparent weight will decrease. However, if the frame of reference is accelerating in the opposite direction, the apparent weight will increase. While a satellite orbits the Earth, apparent weight on the satellite is decreased due to the satellites acceleration. The phenomenon of weightlessness occurs because the frame of reference (the satellite) has a centripetal acceleration caused by gravity. A passenger on the satellite feels weightless, just as he or she would inside a freely falling elevator. Keplers Laws and Newtons SynthesisNewtons work confirmed Keplers laws, which had been developed earlier. Keplers first law states that planetary pathways are ellipt