Practice Problem 0451
The beat frequency is calculated by estimating the number of cycles in a given time. In 12.0 +/-0.2
seconds, 18+/-1 cycles are counted. Calculate
a) The beat frequency
b) The range for beat frequency
c) The absolute error for beat frequency
d) The relative error for beat frequency
e) The relative error for time
f) The relative error for the number of cycles
(see next page for the solution)
a) To determine beat frequency, we use the following equations
bf =
N
Δt
N represents the number of beats, (N = 18) in 12 seconds
bf =
18
12
bf = 1.5 Hz
b) To determine the range, initially we need to calculate the range for each variable
11.8 ≤ Δt ≤ 12.2 seconds
(added/subtracted the absolute error of 0.2 seconds to the measurement of 12 seconds)
17 ≤ N ≤ 19 seconds
(added/subtracted the absolute error of 1 cycle to the measurement of 18 cycles)
bfH =
19
= 1.61 Hz
11.8
1.39 ≤ bf ≤ 1.61 Hz
c) The absolute error for beat frequency
absolute error =
1.61 - 1.39
= 0.11 Hz
2
bfL =
17
= 1.39 Hz
12.2
d) The relative error for beat frequency
relative error =
absolute error 0.11
=
= 0.0733
measurement 1.5
e) The relative error for time
relative error =
absolute error 0.2
=
= 0.0166
measurement 12
f) The relative error for the number of cycles
relative error =
absolute error 1
= = 0.0555
measurement 18
Note the relative error for time added to the relative error for number of cycles is almost equal to the
relative error for beat frequency.
This is a method one can use to verify that their error analysis is correct.