The Physics
Hypertextbook
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 * period of rotation, rpm revolutions per minute, Hz = hertz = revolutions per second
device, event, phenomenon, process frequency ω (rad/s)
planet Venus, the slowest planet (and backward)−5,832.5hour*2.99 × 10−7
planet Earth, the just right planet23.9345hour*7.29 × 10−5
great red spot of Jupiter 10−4 Hz 10−5
analog clock, hour hand12hour*0.000145
planet Jupiter, the fastest planet9.9250hour*0.000176
analog clock, minute hand1hour*0.00175
London eye30min*0.00349
analog clock, second hand1min*0.105
tornado core 0.1 Hz 6
phonograph record, long play (LP)33⅓rpm3.49
phonograph record, extended play (EP)45rpm4.71
phonograph record, disk (1925 standard)78.26rpm8.20
phonograph record, cylinder (1902 standard)160rpm16.8
laserdisc, standard play, PAL/SECAM1,500rpm157
laserdisc, standard play, NTSC1,800rpm188
hard drive, standard (1990~2000)3,600rpm377
hard drive, laptop (ca. 2010)4,500rpm471
hard drive, economy (ca. 2010)5,400rpm565
hard drive, standard (ca. 2010)7,200rpm754
hard drive, premium (ca. 2010)15,000rpm1,570
pulsar, fastest known (PSR J1748–2446ad)716Hz4,500
microgyroscope (fastest "man-made" rotation) 107Hz 108
quark-gluon plasma (fastest rotating fluid) 1020 Hz 1021
Planck angular frequency (absolute limit)√(c5/ℏG)1.86 × 1043