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FORMULAEMECHANICS
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Contents
Articles
Classical Mechanics Formulae 1
Gravitation Formulae 10
Equations for Properties of Matter 14
References
Article Sources and Contributors 16
Article Licenses
License 17
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Classical Mechanics Formulae 1
Classical Mechanics Formulae
Lead Article: Tables of Physics Formulae
This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Classical
Mechanics.
Mass and Inertia
Mass can be considered to be inertial or gravitational.
Inertial mass is the mass associated with the inertia of a body. By Newton's 3rd Law of Motion, the acceleration of a
body is proportional to the force applied to it. Force divided by acceleration is the inertial mass.
Gravitational mass is that mass associated with gravitational attraction. By Newton's Law of Gravity, the
gravitational force exerted by or on a body is proportional to its gravitational mass.
By Einstein's Principle of Equivalence, inertial and gravitational mass are always equal.
Often, masses occur in discrete or continuous distributions. "Discrete mass" and "continuum mass" are notdifferent
concepts, but the physical situation may demand the calculation either as summation (discrete) or integration
(continuous). Centre of mass is notto be confused with centre of gravity (see Gravitation section).
Note the convenient generalisation of mass density through an n-space, since mass density is simply the amount of
mass per unit length, area or volume; there is only a change in dimension number between them.
Quantity (Common
Name/s)
(Common) Symbol/s Defining Equation SI Units Dimension
Mass density of dimension n
( = n-space)
n = 1 for linear mass density,
n = 2 for surface mass
density,
n = 3 for volume mass
density,
etc
linear mass density ,
surface mass density
,
volume mass density
,
no general symbol for
any dimension
n-space mass density:
special cases are:
kg m-n [M][L]-n
Total descrete mass kg m [M][L]
Total continuum mass n-space mass density
special cases are:
kg [M]
Moment of Mass (No common symbol) kg m [M][L]
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Classical Mechanics Formulae 2
Centre of Mass
(Symbols can vary
enourmously)
ith
moment of mass
Centre of mass for a descrete masses
Centre of a mass for a continuum of mass
m [L]
Moment of Inertia (M.O.I.) M.O.I. for Descrete Masses
M.O.I. for a Continuum of Mass
kg m2
s-1
[M][L]2
Mass Tensor Components
Contraction of the tensor with itself yeilds the more familiar
scalar
kg [M]
M.O.I. Tensor Components
2nd-Order Tensor Matrix form
Contraction of the tensor with itself yeilds the more familiar
scalar
kg m2
s-1
[M][L]2
Moment of Inertia Theorems
Often the calculations for the M.O.I. of a body are not easy; fortunatley there are theorems which can simplify the
calculation.
Theorem Nomenclature Equation
Superposition Principle for
M.O.I. about any chosen Axis
= Resultant M.O.I.
Parallel Axis Theorem = Total mass of body
= Perpendicular distance from an axis
through the C.O.M. to another parallel axis
= M.O.I. about the axis through
the C.O.M.
= M.O.I. about the parallel axis
Perpendicular Axis Theorem i, j, k refer to M.O.I. about any three mutually
perpendicular axes:
the sum of M.O.I. about any two is the third.
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Classical Mechanics Formulae 3
Galilean Transforms
The transformation law from one inertial frame (reference frame travelling at constant velocity - including zero) to
another is the Galilean transform. It is only true for classical (Galilei-Newtonian) mechanics.
Unprimed quantites refer to position, velocity and acceleration in one frame F; primed quantites refer to position,
velocity and acceleration in another frame F'moving at velocity V relative to F. Conversely Fmoves at velocity
(V) relative toF'.
Galilean Inertial Frames = Constant relative velocity between
two framesFandF'.
= Position, velocity, acceleration
as measured in frameF.
= Position, velocity, acceleration
as measured in frameF'.
Relative Position
Relative Velocity
Equivalent Accelerations
Laws of Classical MechanicsThe following general approaches to classical mechanics are summarized below in the order of establishment. They
are equivalent formulations, Newton's is very commonly used due to simplicity, but Hamilton's and Lagrange's
equations are more general, and their range can extend into other branches of physics with suitable modifications.
Newton's Formulation (1687)
Force, acceleration, and the momentum rate of change are all equated neatly inNewton's Laws .
1st Law: A zero resultant force acting ON a body BY an external agent causes
zero change in momentum. The effect is a constant momentum vector and therefore
velocity (including zero).
2nd Law: A resultant force acting ON a body BY an external agent causes
change in momentum.
3rd Law: Two bodies i and j mutually exert forces ON each other BY each other,
when in contact.
The 1st law is a special case of the 2nd law. The laws summarized in two
equations (rather than three where one is a corollary). One is an ordinary
differential equation used to summarize the dynamics of the system, the other
is an equivalance between any two agents in the system. Fij
=
force ON body i BY body j, Fij
= force ON body j BY body i.
In applications to a dynamical system of bodies the two equations (effectively)
combine into one. pi= momentum of body i, and F
E=
resultant external force (due to any agent not part of system). Body i does not
exert a force on itself.
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Classical Mechanics Formulae 4
Euler-Lagrange Formulation (1750s)
The generalized coordinates and generalized momenta of any classical
dynamical system satisfy theEuler-Lagrange Equation, which is a set
of (partial) differential equations describing the minimization of the system.
Written as a single equation:
Hamilton's Formulation (1833)
The generalized coordinates and generalized momenta of any classical dynamical
system also satisfyHamilton's equations , which are a set of (partial) differential
equations describing the time development of the system.
The Hamiltonian as a function of generalized coordinates and momenta has the
general form:
The value of the Hamiltonian His the total energy of the dynamical system. For an isolated system, it generally
equals the total kinetic Tand potential energy V.
Hamiltonians can be used to analyze energy changes of many classical systems; as diverse as the simplist
one-body motion to complex many-body systems. They also apply in non-relativistic quantum mechanics; in therelativistic formulation the hamiltonian can be modified to be relativistic like many other quantities.
Derived Kinematic Quantities
For rotation the vectors are axial vectors (also known as pseudovectors), the direction is perpendicular to the plane of
the position vector and tangential direction of rotation, and the sense of rotation is determined by a right hand screw
system.
For the inclusion of the scalar angle of rotational position , it is nessercary to include a normal vector to the
plane containing and defined by the position vector and tangential direction of rotation, so that the vector equations
to hold.
Using the basis vectors for polar coordinates, which are , the unit normal is .
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Classical Mechanics Formulae 5
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Velocitym s
-1[L][T]
-1
Accelerationm s
-2[L][T]
-2
Jerk m s-3 [L][T]-3
Angular Velocityrad s
-1[T]
-1
Angular Accelerationrad s
-2[T]
-2
By vector geometry it can be found that:
and hence the corollary using the above definitions:
Derived Dynamic Quantities
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Momentumkg m s
-1[M][L][T]
-1
ForceN = kg m s
-2[M][L][T]
-2
Impulsekg m s
-1[M][L][T]
-1
Angular Momentum
about a position point
kg m2
s-1
[M][L]2[T]
-1
Total, Spin and Orbital
Angular Momentum
kg m2
s-1
[M][L]2[T]
-1
Moment of a Force
about a position point ,
Torque
N m = kg m2
s-2
[M][L]2[T]
-2
Angular Impulse
No common symbol
kg m2
s-1
[M][L]2[T]
-1
Coefficeint of Restitution
usually
but it is possible that
Dimensionless Dimensionless
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Classical Mechanics Formulae 6
Translational Collisions
For conservation of mass and momentum see Conservation and Continuity Equations.
Description Nomenclature Equation
Completley Inelastic Collision
Inelastic Collision
Elastic Collision
Superelastic/Explosive Collision
General Planar Motion
The plane of motion is considered in a the cartesian x-y plane using basis vectors (i, j), or alternativley the polar
plane containing the (r, ) coordinates using the basis vectors .
For any object moving in any path in a plane, the following are general kinematic and dynamic results
[1]:
Quantity Nomenclature Equation
Position = radial position component
= angular position component
= instantaneous radius of
curvature at on the curve
= unit vector directed to centre of
circle of curvature
Velocity = Instantaneous angular velocity
Acceleration = Instantaneous angular acceleration
Centripetal Force = instananeous mass moment
They can be readily derived by vector geometry and using kinematic/dynamic definitions, and prove to be very
useful. Corollaries of momentum, angular momentum etc can immediatley follow by applying the definitions.
Common special cases are:
the angular components are constant, so these represent equations of motion in a streight line the radial
components i.e. is constant, representing circular motion, so these represent equations of motion in a rotating
path (notneccersarily a circle, osscilations on an arc of a circle are possible) and are both constant, and
, representing uniform circular motion and is constant, representing uniform acceleration in
a streight line
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Classical Mechanics Formulae 7
Mechanical Energy
General Definitions
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Mechanical Work dueto a Resultant Force
J = N m = kg m2
s-2
[M][L]2
[T]-2
Work done ON mechanical
system, Work done BY
J = N m = kg m2
s-2
[M][L]2[T]
-2
Potential EnergyJ = N m = kg m
2s-2
[M][L]2[T]
-2
Mechanical PowerW = J s
-1[M][L]
2[T]
-3
Lagrangian J[M][L]
2[T]
-2
Action J s[M][L]
2[T]
-1
Energy Theorems and Principles
Work-Energy Equations
The change in translational and/or kinetic energy of a body is equal to the work done by a resultant force and/or
torque acting on the body. The force/torque is exerted across a path C, this type of integration is a typical example of
a line integral.
For formulae on energy conservation see Conservation and Continuity Equations.
Theorem/Principle (Common) Equation
Work-Energy Theorem for Translation
Work-Energy Theorem for Rotation
General Work-Energy Theorem
Principle of Least Action
A system always minimizes the action associated with all parts of the system.
Various minimized quantity formulations are:
Maupertuis' Formulation
Euler's Formulation
Lagrangian Formulation
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Classical Mechanics Formulae 8
Potential Energy and Work
Everyconservative force has an associatedpotential energy (often incorrectly termed as "potential", which is related
to energy but notexactly the same quantity):
By following two principles a non-relative value to Ucan be consistently assigned:
Wherever theforce is zero, itspotential energy is defined to be zero as well.
Whenever the force doespositive work,potential energy decreases (becomes more negative), and vice versa.
Useful Derived Equations
Description (Common) Symbols General Vector/Scalar Equation
Kinetic Energy
Angular Kinetic Energy
Total Kinetic Energy
Sum of translational and rotational kinetic energy
Mechanical Work due
to a Resultant Torque
Total work done due to resultant forces and torques
Sum of work due to translational and rotational motion
Elastic Potential Energy
Power transfer by a resultant force
Power transfer by a resultant torque
Total power transfer due to resultant forces and torques
Sum of power transfer due to translational and rotational motion
Transport MechanicsHere is a unit vector normal to the cross-section surface at the cross section considered.
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Classical Mechanics Formulae 9
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Flow Velocity Vector Fieldm s
-1[L][T]
-1
Mass Currentkg s
-1[M][T]
-1
Mass Current Densitykg m
-2
s-1
[M][L]-2
[T]-1
Momentum Currentkg m s
-2[M][L][T]
-2
Momentum Current Densitykg m s
-2[M][L][T]
-2
Damping Parameters, Forces and Torques
Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Spring Constant
(Hooke's Law)
N m-1 [M][T]-2
Damping CoefficientN s m
-1[L][T]
-1
Damping Force N[M][L][T]
-2
Damping Ratio dimensionless dimensionless
Logarithmic decrement
is any amplitude, is the
amplitude n successive peaks
later from , where
dimensionless dimensionless
Torsion ConstantN m rad
-1[M][L]
2[T]
-2
Damping Torque N m[M][L]
2[T]
-2
Rotational Damping CoefficientN m s rad
-1[M][L]
2[T]
-1
References
[1][1] 3000 Solved Problems in Physics, Schaum Series, A. Halpern, Mc Graw Hill, 1988, ISBN 9-780070-257344
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Gravitation Formulae 10
Gravitation Formulae
Lead Article: Tables of Physics Formulae
This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of
Gravitation.
Gravitational Field Definitions
A common misconseption occurs between centre of mass and centre of gravity. They are defined in simalar ways but
are not exactly the same quantity. Centre of mass is the mathematical descrition of placing all the mass in the region
considered to one position, centre of gravity is a real physical quantity, the point of a body where the gravitational
force acts. They are only equal if and only if the external gravitational field is uniform.
Contrary to the strong analogy between (classical) gravitation and electrostatics, there are no "centre of charge" or
"centre of electrostatic attraction" analogues.
Quantity Name (Common) Symbol/s Defining Equation SI Units Dimension
Centre of Gravity
(Symbols can vary
enourmously)
ith
moment of mass
Centre of gravity for a descrete masses
Centre of a gravity for a continuum of mass
m [L]
Standard Gravitation
Parameter of a Mass
N m2
kg-1
[L]3
[T]-2
Gravitational Field, Field
Strength, Potential Gradient,
Acceleration
N kg-1
= m s-2
[L][T]-2
Gravitational Fluxm
3s
-2[L]
3[T]
-2
Absolute Gravitational PotentialJ kg
-1[L]
2[T]
-2
Gravitational Potential
DifferanceJ kg
-1[L]
2[T]
-2
Gravitational Potential Energy J[M][L]
2[T]
-2
Gravitational Torsion FieldHz = s
-1[T]
-1
Gravitational Torsion FluxN m s kg
-1
= m2
s-1
[M]2
[T]-1
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Gravitation Formulae 11
Gravitomagnetic FieldHz = s
-1[T]
-1
Gravitomagnetic FluxN m s kg
-1= m
2
s-1
[M]2
[T]-1
Gravitomagnetic Vector
Potential[1]
m s-1
[M] [T]-1
Gravitational Potential Gradient and Field
Laws of Gravitation
Modern Laws
Gravitomagnetism (GEM) Equations:
In an relativley flat spacetime due to weak gravitational fields (by General Relativity), the following gravitational
analogues of Maxwell's equations can be found, to describe an analogous Gravitomagnetic Field. They are well
established by the theory, but have yet to be verified by experiment[2]
.
Einstein Tensor Field (ETF) Equations
where G
is the Einstien tensor:
GEM Equations
Gravitomagnetic Lorentz Force
Classical Laws
It can be found that Kepler's Laws, though originally discovered from planetary observations (also due to Tycho
Brahe), are true for any central forces.For Kepler's 1st law, the equation is nothing physically fundamental; simply the polar equation of an ellipse where
the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, centred on the central star.
e = elliptic eccentricity
a = elliptic semi-major axes = planet aphelion
b = elliptic semi-minor axes = planet perihelion
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Gravitation Formulae 12
Netown's Law of Gravitational Force
Gauss's Law for Gravitation
Kepler's 1st Law
Planets move in an ellipse, with the star at a focus
Kepler's 2nd Law
Kepler's 3rd Law
Gravitational Fields
The general formula for calculating classical gravitational fields, due to any mass distribution, is found by using
Newtons Law, definition of g, and application of calculus:
Uniform Mass Corolaries
For uniform mass distributions the table below summarizes common cases.
For a massive rotating body (i.e. a planet/star etc), the equation is only true for much less massive bodies (i.e. objects
at the surface) in physical contact with the rotating body. Since this is a classical equation, it is only approximatley
true at any rate.
Superposition Principle for
the Gravitational Field
Gravitational Acceleration
Gravitational Field for
a Rotating (spinning about axis) body
= azimuth angle relative to rotation axis
= unit vector perpendicular to rotation
axis, radial from it
Uniform Gravitational Field, Parabolic Motion = Initail Position
= Initail Velocity
= Time of Flight
Use Constant Acc. Equations to obtain
Point Mass
At a point in a local
array of Point Masses
Line of Mass = Mass
= Length of mass distribution
Spherical Shell = Mass
= Radius
Outside/at Surface
Inside
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Gravitation Formulae 13
Spherical Mass Distribution= Mass
= Radius
Outside/at Surface
Inside
Gravitational Potential Energy of a
Physical Pendulum in a Uniform Field
= seperation between pivot and centre of mass
= length from pivot to centre of gravity
= mass of pendulum
= mass moment of pendulum
Gravitational Torque on a physical
Pendulum in a Uniform Field
For non-uniform fields and mass-moments, applying differentials of the scalar and vector products then integrating
gives the general gravitational torque and potential energy as:
Gravitational Potentials
Potential Energy from gravity
Escape Speed
Orbital Energy
External LinksTables of Physics Formulae
Gravitational Field
Gravitational Induction
Gravitomagnetism
General Relativity
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Gravitation Formulae 14
References
[1][1] Gravitation and Inertia, I. Ciufolini and J.A. Wheeler, Princeton Physics Series, 1995, ISBN 0-691-03323-4
[2][2] Gravitation and Inertia, I. Ciufolini and J.A. Wheeler, Princeton Physics Series, 1995, ISBN 0-691-03323-4
Equations for Properties of MatterLead Article: Tables of Physics Formulae
This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Equations
for Properties of Matter.
Friction
Normal Force
Static Friction, lies tangent to the surface
Kinetic Friction, lies tangent to the surface
Drag Force, tangent to the path
Terminal Velocity
Energy dissipation due to Friction
(sound, heat etc)
Stress and strain
Quantity (Common Name/s) (Common) Symbol/s Definining Equation SI Units Dimension
General Stress
F may be any force applied to area A
Pa = N m-2
[M] [T] [L]-1
General Strain
D = dimension (length, area, volume)
= change in dimension
dimensionless dimensionless
General Modulus of ElasticityPa = N m
-2[M] [T] [L]
-1
Yield Strength/
Ultimate Strength
Young's ModulusPa = N m
-2[M] [T] [L]
-1
Shear ModulusPa = N m
-2[M] [T] [L]
-1
Bulk ModulusPa = N m
-2[M] [T] [L]
-1
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Equations for Properties of Matter 15
Fluid Dynamics
density
pressure
pressure difference
pressure at depth
barometer versus manometer
Pascal's principle
Archimedes' Principle
buoyant force
gravitational force when floating
apparent weight
ideal fluid
equation of continuity constant
Bernoulli's Equationconstant
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Article Sources and Contributors 16
Article Sources and ContributorsClassical Mechanics Formulae Source: http://en.wikiversity.org/w/index.php?oldid=767431 Contributors: Berek, Maschen, Poetlister
Gravitation Formulae Source: http://en.wikiversity.org/w/index.php?oldid=747978 Contributors: Maschen
Equations for Properties of Matter Source: http://en.wikiversity.org/w/index.php?oldid=745710 Contributors: Maschen
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Image Sources, Licenses and Contributors 17
License
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