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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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