Simple Harmonic Motion East Los Angeles （简谐运动东洛杉矶）.pdfVIP

1 of 4 10/1/2004 6.5 HARMONIC MOTION 6.5 HARMONIC MOTION • Simple Harmonic Motion • Damped Oscillatory Motion Simple Harmonic Motion In part A of the figure below, a spring with a weight attached to its free end is in equilibrium (or rest) position. If the weight is pulled down a units and released (part B in the figure) the spring’s elasticity causes the weight to rise a units (a 0) above the equilibrium position, as seen in part C, and then oscillate about the equilibrium position. If friction is neglected, this oscillatory motion is described mathematically by a sinusoid. Other applications of this type of motion include sound, electric current, and electromagnetic waves: To develop a general equation for such motion, consider the figure below. Suppose the point P(x,y) moves around the circle counterclockwise at a uniform angular speed ω. Assume that at time t = 0, P is at (a,0). The angle swept out by ray OP at time t is given by θ ωt . The coordinates of point P at time t are x a cosθ a cosωt y a sinθ a sin ωt 1 2 of 4 10/1/2004 6.5 HARMONIC MOTION As P moves around the circle from the point (a,0) to the point Q(0,y) oscillates back and forth along the y-axis between the points (0,a) and (0,-a). Similarly, the point R(x,0) oscillates back and forth between (a,0) and (-a,0). This oscillatory motion is called simple harmonic motion. 2π The amplitude of the motion is |a| and the period is . The moving points P ω and Q or P and R complete one oscillation or cycle per period. The number of ω cycles per unit of time, called the frequency, is the