For Exercises 53-54, use the model d = a e − c t cos ω t or d = a b k t cos w t to represent damped harmonic motion. A pendulum is pulled π 18 radians to one side and then released. The angular displacement θ follows a pattern of damped harmonic motion with each cycle lasting 2 sec . If the maximum displacement for each cycle decreases by 20 % , find a function that models the angular displacement t sec after being released.
For Exercises 53-54, use the model d = a e − c t cos ω t or d = a b k t cos w t to represent damped harmonic motion. A pendulum is pulled π 18 radians to one side and then released. The angular displacement θ follows a pattern of damped harmonic motion with each cycle lasting 2 sec . If the maximum displacement for each cycle decreases by 20 % , find a function that models the angular displacement t sec after being released.
Solution Summary: The author explains the function that models the angular displacement of the pendulum after time, t, following a pattern of damped harmonic motion.
For Exercises 53-54, use the model
d
=
a
e
−
c
t
cos
ω
t
or
d
=
a
b
k
t
cos
w
t
to represent damped harmonic motion.
A pendulum is pulled
π
18
radians to one side and then released. The angular displacement
θ
follows a pattern of damped harmonic motion with each cycle lasting
2
sec
. If the maximum displacement for each cycle decreases by
20
%
, find a function that models the angular displacement
t
sec
after being released.
"The formula specifies the position of a point P that is moving harmonically on a vertical axis, where t is in seconds and d is in centimeters. Determine the amplitude, period, and frequency, and describe the motion of the point during one complete oscillation (starting at t = 0).
Suppose that a piston is moving straight up and down and that its position at time t sec is s = A cos (2πbt), with A and b positive. The value of A is the amplitude of the motion, and b is the frequency (number of times the piston moves up and down each second). What effect does doubling the frequency have on the piston’s velocity, acceleration, and jerk? (Once you find out, you will know why some machinery breaks when you run it too fast.)
University Calculus: Early Transcendentals (3rd Edition)
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