Rotational Kinematics
Discussion
let's rap
discussion
concept | translation | connection | rotation | |||
---|---|---|---|---|---|---|
base quantities | s, r |
s = | θ × r | θ | ||
coordinate systems | r = |
x î + y ĵ | x = y = r2 = θ = |
r cos θ r sin θ x2 + y2 tan−1 (y/x) |
r = | r r̂ + θ θ̂ |
velocity | v = |
dr/dt | v = | ω × r | ω = | dθ/dt |
acceleration | a = |
dv/dt = d2r/dt2 | a = | α × r − ω2r | α = | dω/dt = d2θ/dt2 |
equations of motion | v = x = v2 = |
v0 + at x0 + v0t + ½at2 v02 + 2a(x − x0) |
ω = θ = ω2 = |
ω0 + αt θ0 + ω0t + ½αt2 ω02 + 2α(θ − θ0) |
the museum of obsolete technology
- constant linear velocity
- movie film
- 24 fps
- audio tape
- cassette: 1⅝ ips
- 8-track: ??
- open reel (reel-to-reel): 15 ips, 7½ips
- data tape
- 9-track: ??
- videotape
- VHS: ??
- Betamax: ??
- open reel (reel-to-reel): ??
- optical disk
- laserdisc, extended play: 18.96 m/s
- CD: 1.2–1.4 m/s
- DVD:
- 3.49 m/s (single layer)
- 3.84 m/s (double layer)
- movie film
- constant angular velocity
- phonograph, cylinder
- 160 rpm
- phonograph, disk
- 78.26 rpm
- 45 rpm
- 33⅓ rpm
- 16⅔ rpm
- video
- laserdisc, standard play: 1800/1500 rpm
- magnetic data storage
- floppy disk: ??
- hard drive: ??
- phonograph, cylinder
frequency | ω (rad/s) | device, event, phenomenon, process | |
---|---|---|---|
−5,832.5 | hour* | 2.99 × 10−7 | planet Venus, the slowest† planet (and backward) |
4.0 | day* | 1.82 × 10−5 | Great Red Spot of Jupiter |
23.9345 | hour* | 7.29 × 10−5 | planet Earth, the just right† planet |
12 | hour* | 0.000145 | analog clock, hour hand |
9.9250 | hour* | 0.000176 | planet Jupiter, the fastest† planet |
1 | hour* | 0.00175 | analog clock, minute hand |
30 | min* | 0.00349 | London eye |
1 | min* | 0.105 | analog clock, second hand |
0.1 | Hz | 0.6 | tornado core |
33⅓ | rpm | 3.49 | phonograph record, long play (LP) |
45 | rpm | 4.71 | phonograph record, extended play (EP) |
78.26 | rpm | 8.20 | phonograph record, disk (1925 standard) |
160 | rpm | 16.8 | phonograph record, cylinder (1902 standard) |
1,500 | rpm | 157 | laserdisc, standard play, PAL/SECAM |
1,800 | rpm | 188 | laserdisc, standard play, NTSC |
3,600 | rpm | 377 | hard drive, standard (1990~2000) |
4,500 | rpm | 471 | hard drive, laptop (ca. 2010) |
5,400 | rpm | 565 | hard drive, economy (ca. 2010) |
7,200 | rpm | 754 | hard drive, standard (ca. 2010) |
15,000 | rpm | 1,570 | hard drive, premium (ca. 2010) |
716 | Hz | 4,500 | pulsar, fastest known (PSR J1748–2446ad) |
107 | Hz | 108 | microgyroscope (fastest "man-made" rotation) |
1020 | Hz | 1021 | quark-gluon plasma (fastest rotating fluid) |
√(c5/ℏG) | 1.86 × 1043 | Planck angular frequency (absolute limit) |