The Physics
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Opus in profectus

# Rotational Kinematics

## Discussion

### let's rap

discussion

Translational and rotational quantities compared
concept translation connection rotation
base quantities
s, r

s =  θ × r θ
coordinate systems
r =

x  + y  x =
y =
r2 =
θ =
r cos θ
r sin θ
x2 + y2
tan−1 (y/x)
r =  r  + 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)
• 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: ??
Selected angular speeds (smallest to largest) * period of rotation,  in terms of angular speed, rpm = revolution per minute, Hz = hertz = revolution per second
frequency ω (rad/s) device, event, phenomenon, process
−5,832.5hour*2.99 × 10−7 planet Venus, the slowest planet (and backward)
4.0 day* 1.82 × 10−5 Great Red Spot of Jupiter
23.9345hour*7.29 × 10−5 planet Earth, the just right planet
12hour*0.000145 analog clock, hour hand
9.9250hour*0.000176 planet Jupiter, the fastest planet
1hour*0.00175 analog clock, minute hand
30min*0.00349 London eye
1min*0.105 analog clock, second hand
33⅓rpm3.49 phonograph record, long play (LP)
45rpm4.71 phonograph record, extended play (EP)
78.26rpm8.20 phonograph record, disk (1925 standard)
160rpm16.8 phonograph record, cylinder (1902 standard)
1,500rpm157 laserdisc, standard play, PAL/SECAM
1,800rpm188 laserdisc, standard play, NTSC
3,600rpm377 hard drive, standard (1990~2000)
4,500rpm471 hard drive, laptop (ca. 2010)
5,400rpm565 hard drive, economy (ca. 2010)
7,200rpm754 hard drive, standard (ca. 2010)
15,000rpm1,570 hard drive, premium (ca. 2010)
716Hz4,500 pulsar, fastest known (PSR J1748–2446ad)
107Hz 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)