Rotational Kinematics

Discussion

discussion

Translational and rotational quantities compared
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)

Selected angular speeds (smallest to largest)
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; 2 rotations per day
9.9250	hour*	0.000176	planet Jupiter, the fastest^† planet
1	hour*	0.00175	analog clock, minute hand; 1 rotation per hour
30	min*	0.00349	London eye
1	min*	0.105	analog clock, second hand; 1 rotation per minute
0.1	Hz	0.6	tornado core
		1	1 radian per second, the SI unit of angular speed
33⅓	rpm	3.49	phonograph record, long play (LP)
45	rpm	4.71	phonograph record, extended play (EP)
60	rpm	6.283	1 rotation per second
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	1570	hard drive, premium (ca. 2010)
716	Hz	4500	fastest^† astronomical object (pulsar J1748–2446ad)
10⁷	Hz	10⁸	fastest^† human-made object (microgyroscope)
10²⁰	Hz	10²¹	fastest^† fluid (quark-gluon plasma)
√(c⁵/ℏG)		1.86 × 10⁴³	theoretical limit (Planck angular frequency)