Formulae Classical Mechanics

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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

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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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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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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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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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


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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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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Mechanical Energy

General Definitions


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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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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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


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

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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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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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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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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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

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