• The motion of objects, which move through curved paths, is the result of the body having components to its velocity, which are two dimensional in nature. The two dimensions of motion are independently analyzed and combined to describe the actual path of the body.
• Independence of the motions is the key to the understanding and simple mathematical analysis of curved motion.
• Symmetry of events is often a useful tool in the analysis of this sort of motion. Here the symmetry is that the path of a body moving through a regular two-dimensional path is the same at the beginning and end of the event.
• Galileo and others, in observing the motions of falling objects from moving platforms, concluded that projectiles and objects dropped from moving platforms have exactly the same characteristics of their paths.
• An object, which is launched and proceeds through space freely falling only under the influence of gravity, is called a projectile. Golf balls, shot arrows, things dropped from moving aircraft, artillery shells, and bullets are all considered to be projectiles. Objects under power such as rockets or airplanes are not projectiles.
• Projectile motion is classified and dealt with in this unit as either launched horizontally or launched upward at an angle.
• The path through which a projectile moves is called its trajectory. The shape of the trajectory of a projectile (neglecting air friction) is a parabola. A projectile displays symmetry in its trajectory if the effects of friction are ignored.
• As the name implies, the motion of a body moving in a circular path is called circular motion. A body does not need to move through a complete circle in order to undergo circular motion. Motion through an arc with a fixed radius also constitutes circular motion.
• A body can move through a horizontal circle at constant speed. This is called uniform circular motion and the body undergoes constant acceleration, even though the speed remains constant. The direction of the velocity vector constantly changes and thus the body is accelerating.
• The acceleration of a body undergoing circular motion is called the centripetal acceleration (ac). Equations: ac = v2/r (where v is the instantaneous, but constant speed, preferably in m/s, and r is the radius of the circle) and ac = 4π2r/T2 (where T is the period of one revolution).
• The force, which causes centripetal acceleration, is called the centripetal force (Fc). The centripetal force is the force, acting toward the center of the circle, responsible for causing the centripetal acceleration. Equations: Fc = mac (where m is the mass of the body) or Fc = m v2/r and Fc = m4π2r/T2. Fighter pilots making high-speed turns experience large centripetal forces which can sometimes cause them to lose consciousness. The centripetal force divided by g (9.8 m/s2) is referred to as g forces. Humans in good condition can tolerate up to 9 g's without physical harm (except for a brief loss of consciousness).
Simple harmonic motion is linear motion in which the acceleration is proportional to the distance from the equilibrium position and is directed toward that position. Examples of harmonic motion are a swinging pendulum and a weight bobbing up and down on a spring.
• Periodic motion occurs when a body repeatedly moves over the same path in equal intervals of time.
• A pendulum is an object suspended so that it can swing back and forth about an axis. The motion of a pendulum is independent of its mass. The motion of a pendulum is independent of its amplitude (in swings of 100 or less?). The period of a pendulum is directly proportional to the square root of its length. The period of a pendulum is inversely proportional to the square root of the local acceleration of gravity (and so is sometimes used to measure local gravity). The pendulum equation:
• Frequency is the number of cycles that an object in periodic motion will complete in a set period of time (preferably seconds). The unit for frequency (ν) is Hertz (Hz), which stands for cycles per second. The reciprocal of frequency is the period (T): T = 1/ν. The period is the number of seconds it takes to complete one complete cycle of periodic motion.
William Dietsch posted & edited 7 April 2007 by D Trapp