Simple Harmonic Motion

By letting figure 3 reference circle for the moving

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Unformatted text preview: By letting Figure 3 Reference circle for the moving particle from Schaum’s Outline of College Physics 10th edition (13) (14) and adding the phase constant to the sin function, we get the following for equation (12). sin (15) 2.2.2 Method 2 For those who have had calculus, you should recall that we can differentiate equation (3) in order to get the velocity. Thus cos sin 2.3 (16) (17) The Acceleration of SHM 2.3.1 Method 1 There are two different methods for finding the acceleration for simple harmonic motion. The first method involves no calculus. To start, recall that the acceleration of an object in uniform circular motion has a magnitude given by (18) where acp represents the centripetal acceleration which as a direction of acceleration that is towards the center of the circular path as shown in Figure 4. Thus when the angular position of the particle at point P is θ, the acceleration vector is at an angle θ below the x-axis, and its x-component is –acp cos θ. Again setting, (19) 3 (20) and adding the phase constant to the sin function, we get the following cos (21) We can combine equations (3) and (21) to yield (22) which is a hallmark of simple harmonic motion: In Simple Harmonic Motion, acceleration is proportional to displacement but opposite in sign, the two quantities are related by square of the angular frequency. the the and the Figure 4 Acceleration vectors of a point P. Thus when the displacement has its greatest positive value, the acceleration has its greatest negative value, and conversely. When the displacement is zero, the acceleration is also zero. 2.3.2 Method 2 For those who have had calculus, then you should recall that we can differentiate equation (15) in order to get the acceleration. Thus sin cos 2.4 (23) (24) The Force Law for Simple Harmonic Motion Once we know how the acceleration of a particle varies with time, we can use Newton’s Second Law to learn what forces must act on a particle to give it that acceleration. If we combine Newton’s Second Law and equation (22), we find, for simple Harmonic Motion (25) (26) (27) This result—a restoring force that is proportional to the displacement but opposite in sign—is familiar. Its...
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