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
| device, phenomena, process, or event | frequency | ω (rad/s) | |
|---|---|---|---|
| planet venus, the slowest planet (and backward) | −5,832.5 | hour* | 2.99 × 10−7 |
| planet earth, the just right planet | 23.9345 | hour* | 7.29 × 10−5 |
| great red spot of jupiter | 10−4 | Hz | 10−5 |
| analog clock, hour hand | 12 | hour* | 0.000145 |
| planet jupiter, the fastest planet | 9.9250 | hour* | 0.000176 |
| analog clock, minute hand | 1 | hour* | 0.00175 |
| london eye | 30 | min* | 0.00349 |
| analog clock, second hand | 1 | min* | 0.105 |
| tornado core | 0.1 | Hz | 6 |
| phonograph record, long play (LP) | 33⅓ | rpm | 3.49 |
| phonograph record, extended play (EP) | 45 | rpm | 4.71 |
| phonograph record, disk (1925 standard) | 78.26 | rpm | 8.20 |
| phonograph record, cylinder (1902 standard) | 160 | rpm | 16.8 |
| laserdisc, standard play, PAL/SECAM | 1,500 | rpm | 157 |
| laserdisc, standard play, NTSC | 1,800 | rpm | 188 |
| hard drive, standard (1990~2000) | 3,600 | rpm | 377 |
| hard drive, laptop (ca. 2010) | 4,500 | rpm | 471 |
| hard drive, economy (ca. 2010) | 5,400 | rpm | 565 |
| hard drive, standard (ca. 2010) | 7,200 | rpm | 754 |
| hard drive, premium (ca. 2010) | 15,000 | rpm | 1,570 |
| pulsar, fastest known (PSR J1748–2446ad) | 716 | Hz | 4,500 |
| microgyroscope (fastest "man-made" rotation) | 107 | Hz | 108 |
| quark-gluon plasma (fastest rotating fluid) | 1020 | Hz | 1021 |
| planck angular frequency (absolute limit) | √(c5/ℏG) | 1.86 × 1043 | |