Physics Formulae/Classical Mechanics Formulae

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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[edit]

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 not different concepts, but the physical situation may demand the calculation either as summation (discrete) or integration (continuous). Centre of mass is not to 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]
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[edit]

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.


Galilean Transforms[edit]

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 F moves at velocity (—V) relative to F' .


Galilean Inertial Frames = Constant relative velocity between

two frames F and F'.

= Position, velocity, acceleration

as measured in frame F .


= Position, velocity, acceleration

as measured in frame F' .

Relative Position


Relative Velocity

Equivalent Accelerations

Laws of Classical Mechanics[edit]

The 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)[edit]

Force, acceleration, and the momentum rate of change are all equated neatly in Newton'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 FE =

resultant external force (due to any agent not part of system). Body i does not

exert a force on itself.

Euler-Lagrange Formulation (1750s)[edit]

The generalized coordinates and generalized momenta of any classical

dynamical system satisfy the Euler-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)[edit]

The generalized coordinates and generalized momenta of any classical dynamical

system also satisfy Hamilton'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 H is the total energy of the dynamical system. For an isolated system, it generally equals the total kinetic T and 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 the relativistic formulation the hamiltonian can be modified to be relativistic like many other quantities.

Derived Kinematic Quantities[edit]

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 .

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Velocity m s-1 [L][T]-1
Acceleration m s-2 [L][T]-2
Jerk m s-3 [L][T]-3
Angular Velocity rad s-1 [T]-1
Angular Acceleration rad s-2 [T]-2


By vector geometry it can be found that:



and hence the corollary using the above definitions:


Derived Dynamic Quantities[edit]

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Momentum kg m s-1 [M][L][T]-1
Force N = kg m s-2 [M][L][T]-2
Impulse kg 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


Translational Collisions[edit]

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[edit]

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, θ) 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 (not neccersarily 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

Mechanical Energy[edit]

General Definitions[edit]

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Mechanical Work due

to 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 Energy J = N m = kg m2 s-2 [M][L]2[T]-2
Mechanical Power W = 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[edit]

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

Potential Energy and Work[edit]

Every conservative force has an associated potential energy (often incorrectly termed as "potential", which is related to energy but not exactly the same quantity):



By following two principles a non-relative value to U can be consistently assigned:

— Wherever the force is zero, its potential energy is defined to be zero as well.

— Whenever the force does positive work, potential energy decreases (becomes more negative), and vice versa.

Useful Derived Equations[edit]

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 Mechanics[edit]

Here is a unit vector normal to the cross-section surface at the cross section considered.


Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Flow Velocity Vector Field m s-1 [L][T]-1
Mass Current kg s-1 [M][T]-1
Mass Current Density kg m-2 s-1 [M][L]-2[T]-1
Momentum Current kg m s-2 [M][L][T]-2
Momentum Current Density kg m s-2 [M][L][T]-2

Damping Parameters, Forces and Torques[edit]

Quantity (Common Name/s) (Common) Symbol/s Defining Equation SI Units Dimension
Spring Constant

(Hooke's Law)

N m-1 [M][T]-2
Damping Coefficient N 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 Constant N m rad-1 [M][L]2[T]-2
Damping Torque N m [M][L]2[T]-2
Rotational Damping Coefficient N m s rad-1 [M][L]2[T]-1


References[edit]

  1. 3000 Solved Problems in Physics, Schaum Series, A. Halpern, Mc Graw Hill, 1988, ISBN 9-780070-257344