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)
23.9345	hour*	7.29 × 10⁻⁵	planet Earth, the just right planet
10⁻⁴	Hz	10⁻⁵	great red spot of Jupiter
12	hour*	0.000145	analog clock, hour hand
9.9250	hour*	0.000176	planet Jupiter, the fastest planet
1	hour*	0.00175	analog clock, minute hand
30	min*	0.00349	London eye
1	min*	0.105	analog clock, second hand
0.1	Hz	6	tornado core
33⅓	rpm	3.49	phonograph record, long play (LP)
45	rpm	4.71	phonograph record, extended play (EP)
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	1,570	hard drive, premium (ca. 2010)
716	Hz	4,500	pulsar, fastest known (PSR J1748–2446ad)
10⁷	Hz	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)