Rotational Kinematics

Discussion

let's rap

discussion

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)
Translational and rotational quantities compared

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, 7½ ips
    • 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: ??
device, phenomena, process, or event value ω (rad/s)
analog clock, hour hand 12 hour 0.000145
analog clock, minute hand 1 hour 0.00175
analog clock, second hand 1 min 0.105
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
laserdisc, standard play, NTSC 1,800 rpm 188
laserdisc, standard play, PAL/SECAM 1,500 rpm 157
london eye 30 min 0.00349
microgyroscope (fastest "man-made" rotation) 10,000,000 Hz* 6.28 × 107
phonograph record, cylinder (1902 standard) 160 rpm 16.8
phonograph record, disk (1925 standard) 78.26 rpm 8.20
phonograph record, extended play (EP) 45 rpm 4.71
phonograph record, long play (LP) 33⅓ rpm 3.49
planet earth, the just right planet 23.9345 hour 7.29 × 10−5
planet jupiter, the fastest planet 9.9250 hour 1.76 × 10−4
planet venus, the slowest planet (and backward) −5,832.5 hour 2.99 × 10−7
planck angular frequency (absolute limit) √(c5/ℏG) 1.86 × 1043
pulsar, fastest known (PSR J1748–2446ad) 716 Hz* 4,500
Selected angular speeds
*rotations per secondperiod of rotation