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The inverse of the period is called the frequency f of the motion. Whereas the pe- riod is the time interval per oscillation, the frequency represents the number of oscil- lations that the particle undergoes per unit time interval: (15.11) The units of f are cycles per second, or hertz (Hz). Rearranging Equation 15.11 gives (15.12) We can use Equations 15.9, 15.10, and 15.11 to express the period and frequency of the motion for the particle–spring system in terms of the characteristics m and k of the (15.13) (15.14) That is, the period and frequency depend only on the mass of the particle and the We can obtain the velocity and acceleration 2 of a particle undergoing simple har- monic motion from Equations 15.7 and 15.8: (15.15) (15.16) From Equation 15.15 we see that, because the sine and cosine functions oscillate between & 1, the extreme values of the velocity v are & 'A. Likewise, Equation 15.16 2 A. Therefore, the maxi- mum values of the magnitudes of the velocity and acceleration are (15.17) (15.18) Figure 15.6a plots position versus time for an arbitrary value of the phase constant. The associated velocity–time and acceleration–time curves are illustrated in Figures x is a maximum or a minimum, the velocity is zero. Likewise, when x is zero, the speed is a maximum. Furthermore, note that the a
max ! ' 2 A ! k m
A v max ! ' A ! √ k m
A a ! d
2 x dt
2 ! " ' 2 A cos('t % () v ! dx dt ! " ' A sin('t % () f ! 1 T ! 1 2)
√ k m T ! 2) ' ! 2)
√ m k ' ! 2)f ! 2) T f ! 1 T ! ' 2) S E C T I O N 15 . 2 • Mathematical Representation of Simple Harmonic Motion 457 2 Because the motion of a simple harmonic oscillator takes place in one dimension, we will denote velocity as v and acceleration as a, with the direction indicated by a positive or negative sign, as in Maximum magnitudes of speed and acceleration in simple harmonic motion Acceleration of an object in simple harmonic motion Velocity of an object in simple harmonic motion Period Frequency ▲ PITFALL PREVENTION 15.4 Two Kinds of Frequency We identify two kinds of fre- , the angular frequency, is mea- sured in radians per second. Be |