Value 
Equation 
Notes 
Units 
Angle 
θ = 2πt / T
= ωt
= s / r 
swept angle = 2π * time / period
= angular velocity * time
= displacement / radius 
rad = s · s^{1}
= rad·s^{1} · s
= m · m^{1} 
Angular frequency 
ω = dθ / dt
= 2π / T
= 2πf
= v / r 
angular velocity = change in angle over change in time = 2π / period = 2π * frequency = linear velocity / radius
This is the
differential of angle w.r.t. time.
Also called angular speed, and angular velocity (although that one is a little different, being a pseudovector). 
rad·s^{1} = rad per turn · s^{1}
= rad per turn · Hz
= m·s^{1} · m^{1} 
Angular velocity of a spring in SHM 
ω = √ (k / m) 
angular velocity = square root of (spring constant / mass)

rad·s^{1} = (N·m^{1} · kg^{1})^{½} 
Angular acceleration 
α = dω / dt 
angular acceleration = charge in angular velocity over change in time
This is the
differential of angular velocity w.r.t. time. 
rad·s^{2} 
Angular momentum 
L = Iω 
angular momentum = rotational inertia * angular frequency 
kg·m^{2}·s^{1} = kg·m^{2} · rad·s^{1} 
Angular momentum of thin shell cylinder 
L = mr^{2}ω 
angular momentum = mass * radius squared * angular frequency 
kg·m^{2}·s^{1} = kg · m^{2} · rad·s^{1} 
Centripetal force 
F = mv^{2} / r 
Centripetal force (centrifugal force) = mass * velocity squared / radius
This is simply F = ma with the formula for linear acceleration from below substituted in. 
N^{} = kg · (m·s^{1})^{2} · m^{1} 
Energy 
E = τθ 
energy = torque * angle
Hence torque can be given in joules per radian.
Dividing both sides by time gives power (see below). 
J = N·m * rad 
Frequency 
f = 1 / T 
frequency = 1 / period 
Hz^{} = s^{1} 
Hooke's Law
(force on a spring) 
F = kX 
restoring force (hence the negative sign) is proportional to the displacement from equilibrium
Can also be stated without the negative sign, as the force required to produce the displacement. Each spring has its own value of k, the spring constant. 
N = N·m^{1} · m 
Linear velocity 
v = 2πr / T
= rω 
linear velocity = 2π * radius / period = radius * angular velocity
Also called tangential velocity. 
m·s^{1} = m · rad·s^{1} 
Linear acceleration 
a = v^{2} / r 
linear acceleration = velocity squared / radius 
m·s^{2} = (m·s^{1})^{2} · m^{1} 
Moments in equilibrium 
F_{1}d_{1} + F_{2}d_{2} = F_{3}d_{3} + F_{4}d_{4} 
sum of anticlockwise moments = sum of clockwise moments 
N · m = N · m 
Pendulm 
T = 2π√(L / g) 
period = 2π * square root of (pendulum length / gravitational acceleration)
For very small swing angles. 
s = [const.] * √(m / m·s^{2}) 
Period 
T = 1 / f 
period = 1 / frequency 
s^{1} = Hz 
Power 
P = τω
E / t = τθ / t 
power = torque * angular velocity
This is the scalar product of two vectors. Throw in 2π / 60 if using rpm, such as for a car engine. 
W = N·m · rad·s^{1} 
Rotational inertia 
I = L / ω 
rotational inertia = angular momentum / angular frequency
Rotational inertia is also called moment of inertia. 
kg · m^{2} = kg·m^{2}·s^{1} / rad·s^{1} 
Rotational inertia of thin shell cylinder or pendulum 
I = mr^{2} 
rotational inertia = mass * radius (from pivot for pendulum) squared 
kg·m^{2} = kg · m^{2} 
... of solid cylinder 
I = ½mr^{2} 
rotational inertia = half * mass * radius squared 
kg·m^{2} = [const.] · kg · m^{2} 
... of thin shell sphere 
I = ^{2}/_{3} mr^{2} 
rotational inertia = two thirds * mass * radius squared 
kg·m^{2} = [const.] · kg · m^{2} 
... of solid sphere 
I = ^{2}/_{5} mr^{2} 
rotational inertia = two thirds * mass * radius squared 
kg·m^{2} = [const.] · kg · m^{2} 
... of long rod pivoted at centre 
I = ^{1}/_{12} mL^{2} 
rotational inertia = one twelfth * mass * radius squared 
kg·m^{2} = [const.] · kg · m^{2} 
... of long rod pivoted at one end 
I = ^{1}/_{3} mL^{2} 
rotational inertia = one third * mass * radius squared 
kg·m^{2} = [const.] · kg · m^{2} 
Rotational kinetic energy 
E_{k} = ½Iω^{2}

kinetic energy = half * angular momentum * angular frequency squared 
J = [const.] · kg · (m·s^{1})^{2} 
SHM displacement 
x = x_{0} sin ^{}ωt 
displacement = amplitude * sine( angular frequency * time )
Sine can be replaced with cosine (depending on start position of SHM). 
m = m · [ratio] 
SHM velocity 
v = ωx_{0} cos ωt
= ±ω √( x_{0}^{2}  x^{2} ) 
velocity = angular frequency * amplitude * cosine( angular frequency * time )
Cosine can be replaced with sine (depending on start position of SHM). 
m·s^{1} = rad·s^{1} · m · [ratio] 
SHM acceleration 
a = –ω^{2}x
= –ω^{2}x_{0} sin ^{}ωt ^{}
= –4π^{2}f^{2}x ^{} 
acceleration = negative angular velocity squared * dispacement
The negative is because the acceleration is always in a direction opposite to the displacement from equilibrium.
Sine can be replaced with cosine (depending on start position of SHM).
This is closely related to Hooke's Law. 
m·s^{2} = (rad·s^{1})^{2} · m 
Torque 
τ = Fd 
torque = force * pependicular distance 
N·m = N · m 
Torque 
τ = Iα 
torque = rotational inertia * angular acceleration
Assumes the body is not changing size, which would change the rotational inertia (see above). 
N·m = kg·m^{2} · rad·s^{2} 