Oscillation as Rotation. Rate of rotation: ω degrees/sec
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1 Oscillation as Rotation Rate of rotation: ω degrees/sec
2 Oscillation as Rotation Rate of rotation: ω degrees/sec
3 Oscillation as Rotation
4 Oscillation as Rotation y x
5 Oscillation as Rotation y R x
6 Oscillation as Rotation y R x
7 Oscillation as Rotation y R x
8 Oscillation as Rotation y R θ x
9 Oscillation as Rotation y R θ x
10 Oscillation as Rotation y R θ x
11 Oscillation as Rotation y R θ x x
12 Oscillation as Rotation y R x = R cosθ θ x x
13 Oscillation as Rotation y R x = R cosθ θ x x θ = ωt
14 Oscillation as Rotation y R x = R cosθ θ x x θ = ωt ω is the angular velocity, e.g. in degrees/sec
15 Oscillation as Rotation y R x = R cosθ θ x x θ = ωt ω is the angular velocity, e.g. in degrees/sec For simplicity, we will measure it in degrees per discrete tick or sample. (Think of a clock with a digital second hand and rotates with discrete jumps).
16 x as a function of θ (R=1)
17 x as a function of θ (R=1) x = R cosθ
18 x!projection x as a function of θ (R=1) x = R cosθ ω= 45 /tick !0.5!1! !0.5! Theta in degrees
19 x!projection x as a function of θ (R=1) x = R cosθ ω= 45 /tick ω= 30 /tick !0.5!1!1 0 1! "#$ "!"#$!!!! "! 1! 0.5 0!0.5 6!7489,:-081 "#$ "!"#$! Theta in degrees!! "! "" %"" & "" '"" $ "" ("" )"" *+,-./01/2,3 4,,5
20 x as a function of t
21 x as a function of t x = R cosθ
22 x as a function of t value of x as a function of time is determined by how θ changes with time. x = R cosθ
23 x as a function of t value of x as a function of time is determined by how θ changes with time. x = R cosθ θ = ω t
24 x as a function of t value of x as a function of time is determined by how θ changes with time. x = R cosθ θ = ω t degrees/sec sec
25 x as a function of t value of x as a function of time is determined by how θ changes with time. But if we measure ω in degrees per tick, the time function will depend also on how long (in seconds) each tick or digital sample is. x = R cosθ θ = ω t degrees/sec sec
26 x as a function of t value of x as a function of time is determined by how θ changes with time. But if we measure ω in degrees per tick, the time function will depend also on how long (in seconds) each tick or digital sample is. x = R cosθ θ = ω t degrees/sec sec τ = tick duration in secs
27 x as a function of t value of x as a function of time is determined by how θ changes with time. But if we measure ω in degrees per tick, the time function will depend also on how long (in seconds) each tick or digital sample is. x = R cosθ θ = ω t degrees/sec sec τ = tick duration in secs s = 1/τ ticks (samples)/sec
28 x as a function of t value of x as a function of time is determined by how θ changes with time. But if we measure ω in degrees per tick, the time function will depend also on how long (in seconds) each tick or digital sample is. x = R cosθ θ = ω t degrees/sec sec θ = ω s t τ = tick duration in secs s = 1/τ ticks (samples)/sec
29 x as a function of t value of x as a function of time is determined by how θ changes with time. But if we measure ω in degrees per tick, the time function will depend also on how long (in seconds) each tick or digital sample is. x = R cosθ θ = ω t degrees/sec sec θ = ω s t degrees/tick ticks/sec sec τ = tick duration in secs s = 1/τ ticks (samples)/sec
30 x as a function of t
31 x as a function of t! s=1 "#$ "!"#$!!!! "!! 1!23.4)-5'.+ "#$ "!"#$!! " $!"!$ %" %$ &'()*'+*,)-.+/0
32 x as a function of t! s=1 "#$ "!"#$!!!! "!! "#$ "!"#$ s=2 1!23.4)-5'.+! "#$ "!"#$!! " $!"!$ %" %$ &'()*'+*,)-.+/0!!!! "!! 4!5617,08*1. "#$ "!"#$!! " % & ' (!"!% )*+,-*.-/,01.23
33 x as a function of t! s=1 "#$ "!"#$!!!! "!! "#$ "!"#$ s=2!!!! "! 1!23.4)-5'.+! "#$ "!"#$!! " $!"!$ %" %$ &'()*'+*,)-.+/0! "#$ "!"#$ s=10000!!!! "! 4!5617,08*1.! "#$ "!"#$!! " % & ' (!"!% )*+,-*.-/,01.23 &!3405,/6*0-! "#$ "!"#$!! " "#$!!#$ % %#$ )*+,'*-'.,/0-12 &'!"!(
34 ω and f f = ω 360 s
35 ω and f What is the relation between ω in degrees/ tick and f (frequency) in cycles/sec (Hz)? f = ω 360 s
36 ω and f What is the relation between ω in degrees/ tick and f (frequency) in cycles/sec (Hz)? There are 360º in a cycle, so to get cycles/ tick, divide by 360: f = ω 360 s
37 ω and f What is the relation between ω in degrees/ tick and f (frequency) in cycles/sec (Hz)? There are 360º in a cycle, so to get cycles/ tick, divide by 360: ω/360 f = ω 360 s
38 ω and f What is the relation between ω in degrees/ tick and f (frequency) in cycles/sec (Hz)? There are 360º in a cycle, so to get cycles/ tick, divide by 360: ω/360 To get cycles per second, multiply cycles/tick by ticks/second (sampling rate). So: f = ω 360 s
39 Examples
40 Examples ω=45, s=1, f=1/8 Hz
41 Examples ω=45, s=1, f=1/8 Hz ω=45, s=2, f=1/4 Hz
42 Examples ω=45, s=1, f=1/8 Hz ω=45, s=2, f=1/4 Hz ω=45, s=10000, f=1250 Hz
43 Radian measure of angle y π/4 x
44 Radian measure of angle Circumference of a circle C = 2πr. y π/4 x
45 Radian measure of angle Circumference of a circle C = 2πr. y So for a circle with r=1, C=2π. π/4 x
46 Radian measure of angle Circumference of a circle C = 2πr. y So for a circle with r=1, C=2π. A 360º angle will have a path length of 2π. π/4 x
47 Radian measure of angle Circumference of a circle C = 2πr. y So for a circle with r=1, C=2π. A 360º angle will have a path length of 2π. π/4 x So a 45º angle, which is 1/8 of a circle, will have a path length of π/4.
48 Radian measure of angle Circumference of a circle C = 2πr. y So for a circle with r=1, C=2π. A 360º angle will have a path length of 2π. π/4 x So a 45º angle, which is 1/8 of a circle, will have a path length of π/4.
49 Radian measure of angle Circumference of a circle C = 2πr. y So for a circle with r=1, C=2π. A 360º angle will have a path length of 2π. θ π/4 x So a 45º angle, which is 1/8 of a circle, will have a path length of π/4.
50 Radian measure of angle y x
51 Radian measure of angle Circles with different radii will have different circumferences. y x
52 Radian measure of angle Circles with different radii will have different circumferences. y But a given angle definition should not depend on the actual circumference. x
53 Radian measure of angle Circles with different radii will have different circumferences. y But a given angle definition should not depend on the actual circumference. x
54 Radian measure of angle Circles with different radii will have different circumferences. y But a given angle definition should not depend on the actual circumference. θ x
55 Radian measure of angle Circles with different radii will have different circumferences. y But a given angle definition should not depend on the actual circumference. θ x So we normalize C by dividing by r, meaning that a complete circle is always 2π radians, 45º is always π/4 radians, etc.
56 ω (in radians/tick) to f
57 ω (in radians/tick) to f There are 2π radians in a cycle, so to get cycles/tick, divide by 2π:
58 ω (in radians/tick) to f There are 2π radians in a cycle, so to get cycles/tick, divide by 2π: To get cycles per second, multiply cycles/tick by ticks/second (sampling rate). So:
59 ω (in radians/tick) to f There are 2π radians in a cycle, so to get cycles/tick, divide by 2π: To get cycles per second, multiply cycles/tick by ticks/second (sampling rate). So:
60 ω (in radians/tick) to f There are 2π radians in a cycle, so to get cycles/tick, divide by 2π: To get cycles per second, multiply cycles/tick by ticks/second (sampling rate). So: f = ω 2π s
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