Symbol

Greek alphabet table
Greek Symbol
Greek Letter
Name

English
Equivalent

Upper
Case

Lower
Case

Α

α

Alpha

a

Β

β

Beta

b

Γ

γ

Gamma

g

Γ

δ

Delta

d

Δ

ε

Epsilon

e

Ε

δ

Zeta

z

Ζ

ε

Eta

h

Θ

ζ

Theta

th

Η

η

Iota

i

Κ

θ

Kappa

k

Λ

ι

Lambda

l

Μ

κ

Mu

m

Ν

λ

Nu

n

Ξ

μ

Xi

x

Ο

ν

Omicron

o

Π

π

Pi

p

Ρ

ξ

Rho

r

΢

ζ

Sigma

s

Σ

η

Tau

t

Τ

υ

Upsilon

u

Φ

θ

Phi

ph

Υ

χ

Chi

ch

Φ

ψ

Psi

ps

Χ

ω

Omega

o

Angular Velocity
For an object rotating about an axis,
every point on the object has the same
angular velocity. The tangential
velocity of any point is proportional
to its distance from the axis of
rotation. Angular velocity has the
units rad/s.

Angular velocity is the rate of change
of angular displacement and can be
described by the relationship

Angular velocity can be considered to be a vector
quantity, with direction along the axis of rotation in
theright-hand rule sense.
Vector angular velocity

and if v is constant, the angle can be
calculated from

Angular Momentum of a Particle
The angular momentum
of a particle of mass m
with respect to a chosen
origin is given by

L = mvr sin θ
or more formally by
the vector product
Index

L=rxp
The direction is given
by the right hand
rule which would give
L the direction out of
the diagram. For an
orbit, angular
momentum
is conserved, and this
leads to one of Kepler's
laws. For a circular
orbit, L becomes

Angular momentum of
rigid body

L = mvr
Go Back
HyperPhysics***** Mechanics *****Rotational motion

R Nave

Angular Momentum
The angular momentum of a rigid object is defined as the product of the moment of inertia and
theangular velocity. It is analogous to linear momentum and is subject to the fundamental constraints of
the conservation of angular momentum principle if there is no external torque on the object. Angular
momentum is a vector quantity. It is derivable from the expression for the angular momentum of a
particle

Index

Rotational-Linear Parallels

Moment
of inertia
concepts

Go Back
HyperPhysics***** Mechanics ***** Rotation

R Nave

Moment of Inertia
Moment of inertia is the name given to rotational inertia, the rotational analog
of massfor linear motion. It appears in the relationships for the dynamics of
rotational motion. The moment of inertia must be specified with respect to a
chosen axis of rotation. For a point mass the moment of inertia is just the mass
times the square of perpendicular distance to the rotation axis, I = mr2. That
point mass relationship becomes the basis for all other moments of inertia since
any object can be built up from a collection of point masses.

Index
Moment
of
inertia
concepts

Common forms Examples General form Development for point mass

Go Back
HyperPhysics***** Mechanics ***** Rotation

R Nave

Common Moments of Inertia

Index
Moment
of inertia
concepts

Go Back
HyperPhysics***** Mechanics ***** Rotation

R Nave

Moment of Inertia Examples
Moment of inertia is defined with respect to a specific rotation axis. The
moment of inertia of a point mass with respect to an axis is defined as the
product of the mass times the distance from the axis squared. The moment of
inertia of any extended object is built up from that basic definition. The general
form of the moment of inertia involves an integral.

Index
Moment
of
inertia
concepts

Moments of inertia for common forms
Where moment of inertia appears in physical quantities

Go Back
HyperPhysics***** Mechanics ***** Rotation

R Nave

Moment of Inertia, General Form
Since the moment of inertia of an ordinary object involves a continuous distribution of
mass at a continually varying distance from any rotation axis, the calculation of
moments of inertia generally involves calculus, the discipline of mathematics which can
handle such continuous variables. Since the moment of inertia of a point massis defined
by

then the moment of inertia contribution by an infinitesmal mass element dm has the
same form. This kind of mass element is called a differential element of mass and its
moment of inertia is given by

Note that the differential element of moment of inertia dI must always be defined with
respect to a specific rotation axis. The sum over all these mass elements is called
an integral over the mass.

Usually, the mass element dm will be expressed in terms of the geometry of the object,
so that the integration can be carried out over the object as a whole (for example, over a
long uniform rod).
Having called this a general form, it is probably appropriate to point out that it is a
general form only for axes which may be called "principal axes", a term which includes
all axes of symmetry of objects. The concept of moment of inertia for general objects
about arbitrary axes is a much more complicated subject. The moment of inertia in such
cases takes the form of a mathematical tensor quantity which requires nine components
to completely define it.

Radius Of Gyration

Sometimes the moment of inertia of a body about a specified axis will be given using the
radius of gyration, k. Given the value k and the mass of the body (m) we can calculate its
moment of inertia (I), using the following formula:

Consequently, the radius of gyration is given by

The physical interpretation of the radius of gyration is that it is the radius of a uniform thin
hoop (or ring), having the same moment of inertia (about an axis passing through its
geometric center — shown below), as the given body about the specified axis.

The moment of inertia of the thin hoop about the blue axis passing through its geometric
center is:

The radius of gyration can be useful for listing in a table. If you want to know the moment
of inertia of a complex shaped body about a given axis you simply look up its radius of
gyration, and then (knowing its mass) apply the above formula to find the moment of

inertia.