World Library  
Flag as Inappropriate
Email this Article

List of equations in classical mechanics

Article Id: WHEBN0000006050
Reproduction Date:

Title: List of equations in classical mechanics  
Author: World Heritage Encyclopedia
Language: English
Subject: Classical mechanics, List of equations in gravitation, List of equations in wave theory, List of photonics equations, Defining equation (physics)
Collection: Classical Mechanics, Equations of Physics, Mathematics-Related Lists
Publisher: World Heritage Encyclopedia
Publication
Date:
 

List of equations in classical mechanics

Classical mechanics is the branch of physics used to describe the motion of macroscopic objects.[1] It is the most familiar of the theories of physics. The concepts it covers, such as mass, acceleration, and force, are commonly used and known.[2] The subject is based upon a three-dimensional Euclidean space with fixed axes, called a frame of reference. The point of concurrency of the three axes is known as the origin of the particular space.[3]

Classical mechanics utilises many equations—as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equations, manifolds, Lie groups, and ergodic theory.[4] This page gives a summary of the most important of these.

This article lists equations from Newtonian mechanics, see analytical mechanics for the more general formulation of classical mechanics (which includes Lagrangian and Hamiltonian mechanics).

Contents

  • Classical mechanics 1
    • Mass and inertia 1.1
    • Derived kinematic quantities 1.2
    • Derived dynamic quantities 1.3
    • General energy definitions 1.4
    • Generalized mechanics 1.5
  • Kinematics 2
  • Dynamics 3
    • Precession 3.1
  • Energy 4
  • Euler's equations for rigid body dynamics 5
  • General planar motion 6
    • Central force motion 6.1
  • Equations of motion (constant acceleration) 7
  • Galilean frame transforms 8
  • Mechanical oscillators 9
  • See also 10
  • Notes 11
  • References 12

Classical mechanics

Mass and inertia

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Linear, surface, volumetric mass density λ or μ (especially in acoustics, see below) for Linear, σ for surface, ρ for volume. m = \int \lambda \mathrm{d} \ell

m = \iint \sigma \mathrm{d} S

m = \iiint \rho \mathrm{d} V \,\!

kg mn, n = 1, 2, 3 [M][L]n
Moment of mass[5] m (No common symbol) Point mass:

\mathbf{m} = \mathbf{r}m \,\!

Discrete masses about an axis x_i \,\!:
\mathbf{m} = \sum_{i=1}^N \mathbf{r}_\mathrm{i} m_i \,\!

Continuum of mass about an axis x_i \,\!:
\mathbf{m} = \int \rho \left ( \mathbf{r} \right ) x_i \mathrm{d} \mathbf{r} \,\!

kg m [M][L]
Centre of mass rcom (Symbols vary) ith moment of mass \mathbf{m}_\mathrm{i} = \mathbf{r}_\mathrm{i} m_i \,\!

Discrete masses:
\mathbf{r}_\mathrm{com} = \frac{1}{M}\sum_i \mathbf{r}_\mathrm{i} m_i = \frac{1}{M}\sum_i \mathbf{m}_\mathrm{i} \,\!

Mass continuum:
\mathbf{r}_\mathrm{com} = \frac{1}{M}\int \mathrm{d}\mathbf{m} = \frac{1}{M}\int \mathbf{r} \mathrm{d}m = \frac{1}{M}\int \mathbf{r} \rho \mathrm{d}V \,\!

m [L]
2-Body reduced mass m12, μ Pair of masses = m1 and m2 \mu = \left (m_1m_2 \right )/\left ( m_1 + m_2 \right) \,\! kg [M]
Moment of inertia (MOI) I Discrete Masses:

I = \sum_i \mathbf{m}_\mathrm{i} \cdot \mathbf{r}_\mathrm{i} = \sum_i \left | \mathbf{r}_\mathrm{i} \right | ^2 m \,\!

Mass continuum:
I = \int \left | \mathbf{r} \right | ^2 \mathrm{d} m = \int \mathbf{r} \cdot \mathrm{d} \mathbf{m} = \int \left | \mathbf{r} \right | ^2 \rho \mathrm{d}V \,\!

kg m2 [M][L]2

Derived kinematic quantities

Kinematic quantities of a classical particle: mass m, position r, velocity v, acceleration a.
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Velocity v \mathbf{v} = \mathrm{d} \mathbf{r}/\mathrm{d} t \,\! m s−1 [L][T]−1
Acceleration a \mathbf{a} = \mathrm{d} \mathbf{v}/\mathrm{d} t = \mathrm{d}^2 \mathbf{r}/\mathrm{d} t^2 \,\! m s−2 [L][T]−2
Jerk j \mathbf{j} = \mathrm{d} \mathbf{a}/\mathrm{d} t = \mathrm{d}^3 \mathbf{r}/\mathrm{d} t^3 \,\! m s−3 [L][T]−3
Angular velocity ω \boldsymbol{\omega} = \mathbf{\hat{n}} \left ( \mathrm{d} \theta /\mathrm{d} t \right ) \,\! rad s−1 [T]−1
Angular Acceleration α \boldsymbol{\alpha} = \mathrm{d} \boldsymbol{\omega}/\mathrm{d} t = \mathbf{\hat{n}} \left ( \mathrm{d}^2 \theta / \mathrm{d} t^2 \right ) \,\! rad s−2 [T]−2

Derived dynamic quantities

Angular momenta of a classical object.

Left: intrinsic "spin" angular momentum S is really orbital angular momentum of the object at every point,

right: extrinsic orbital angular momentum L about an axis,

top: the moment of inertia tensor I and angular velocity ω (L is not always parallel to ω)[6]

bottom: momentum p and it's radial position r from the axis.

The total angular momentum (spin + orbital) is J.
Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Momentum p \mathbf{p}=m\mathbf{v} \,\! kg m s−1 [M][L][T]−1
Force F \mathbf{F} = \mathrm{d} \mathbf{p}/\mathrm{d} t \,\! N = kg m s−2 [M][L][T]−2
Impulse J, Δp, I \mathbf{J} = \Delta \mathbf{p} = \int_{t_1}^{t_2} \mathbf{F}\mathrm{d} t \,\! kg m s−1 [M][L][T]−1
Angular momentum about a position point r0, L, J, S \mathbf{L} = \left ( \mathbf{r} - \mathbf{r}_0 \right ) \times \mathbf{p} \,\! Most of the time we can set r0 = 0 if particles are orbiting about axes intersecting at a common point. kg m2 s−1 [M][L]2[T]−1
Moment of a force about a position point r0,

Torque

τ, M \boldsymbol{\tau} = \left ( \mathbf{r} - \mathbf{r}_0 \right ) \times \mathbf{F} = \mathrm{d} \mathbf{L}/\mathrm{d} t \,\! N m = kg m2 s−2 [M][L]2[T]−2
Angular impulse ΔL (no common symbol) \Delta \mathbf{L} = \int_{t_1}^{t_2} \boldsymbol{\tau}\mathrm{d} t \,\! kg m2 s−1 [M][L]2[T]−1

General energy definitions

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Mechanical work due

to a Resultant Force

W W = \int_C \mathbf{F} \cdot \mathrm{d} \mathbf{r} \,\! J = N m = kg m2 s−2 [M][L]2[T]−2
Work done ON mechanical

system, Work done BY

WON, WBY \Delta W_\mathrm{ON} = - \Delta W_\mathrm{BY} \,\! J = N m = kg m2 s−2 [M][L]2[T]−2
Potential energy φ, Φ, U, V, Ep \Delta W = - \Delta V \,\! J = N m = kg m2 s−2 [M][L]2[T]−2
Mechanical power P P = \mathrm{d}E/\mathrm{d}t \,\! W = J s−1 [M][L]2[T]−3

Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U:

  • Wherever the force is zero, its potential energy is defined to be zero as well.
  • Whenever the force does work, potential energy is lost.

Generalized mechanics

Quantity (common name/s) (Common) symbol/s Defining equation SI units Dimension
Generalized coordinates q, Q varies with choice varies with choice
Generalized velocities \dot{q},\dot{Q} \,\! \dot{q}\equiv \mathrm{d}q/\mathrm{d}t \,\! varies with choice varies with choice
Generalized momenta p, P p = \partial L /\partial \dot{q} \,\! varies with choice varies with choice
Lagrangian L L(\mathbf{q},\mathbf{\dot{q}},t) = T(\mathbf{\dot{q}})-V(\mathbf{q},\mathbf{\dot{q}},t) \,\!

where \mathbf{q}=\mathbf{q}(t) \,\! and p = p(t) are vectors of the generalized coords and momenta, as functions of time

J [M][L]2[T]−2
Hamiltonian H H(\mathbf{p},\mathbf{q},t) = \mathbf{p}\cdot\mathbf{\dot{q}} - L(\mathbf{q},\mathbf{\dot{q}},t) \,\! J [M][L]2[T]−2
Action, Hamilton's principal function S, \scriptstyle{\mathcal{S}} \,\! \mathcal{S} = \int_{t_1}^{t_2} L(\mathbf{q},\mathbf{\dot{q}},t) \mathrm{d}t \,\! J s [M][L]2[T]−1

Kinematics

In the following rotational definitions, the angle can be any angle about the specified axis of rotation. It is customary to use θ, but this does not have to be the polar angle used in polar coordinate systems. The unit axial vector

\bold{\hat{n}} = \bold{\hat{e}}_r\times\bold{\hat{e}}_\theta \,\!

defines the axis of rotation, \scriptstyle \bold{\hat{e}}_r \,\! = unit vector in direction of r, \scriptstyle \bold{\hat{e}}_\theta \,\! = unit vector tangential to the angle.

Translation Rotation
Velocity Average:
\mathbf{v}_{\mathrm{average}} = {\Delta \mathbf{r} \over \Delta t}

Instantaneous:

\mathbf{v} = {d\mathbf{r} \over dt}
Angular velocity
\boldsymbol{\omega} = \bold{\hat{n}}\frac = \frac{\Delta\mathbf{v}}{\Delta t}

Instantaneous:

\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2}
Angular acceleration
\boldsymbol{\alpha} = \frac\frac = \frac{\Delta\mathbf{a}}{\Delta t}

Instantaneous:

\mathbf{j} = \frac{d\mathbf{a}}{dt} = \frac{d^2\mathbf{v}}{dt^2} = \frac{d^3\mathbf{r}}{dt^3}
Angular jerk
\boldsymbol{\zeta} = \frac\frac\frac{dt} = \frac{d(m\mathbf{v})}{dt} \\ & = m\mathbf{a} + \mathbf{v}\frac{dt} = \frac{d^2\mathbf{p}}{dt^2} = \frac{d^2(m\mathbf{v})}{dt^2} \\ & = m\mathbf{j} + \mathbf{2a}\frac

where w is the weight of the spinning flywheel.

Energy

The mechanical work done by an external agent on a system is equal to the change in kinetic energy of the system:

General work-energy theorem (translation and rotation)

The work done W by an external agent which exerts a force F (at r) and torque τ on an object along a curved path C is:

W = \Delta T = \int_C \left ( \mathbf{F} \cdot \mathrm{d} \mathbf{r} + \boldsymbol{\tau} \cdot \mathbf{n} {\mathrm{d} \theta} \right ) \,\!

where θ is the angle of rotation about an axis defined by a unit vector n.

Kinetic energy
\Delta E_k = W = \frac{1}{2} m(v^2 - {v_0}^2)
Elastic potential energy

For a stretched spring fixed at one end obeying Hooke's law:

\Delta E_p = \frac{1}{2} k(r_2-r_1)^2 \,\!

where r2 and r1 are collinear coordinates of the free end of the spring, in the direction of the extension/compression, and k is the spring constant.

Euler's equations for rigid body dynamics

Euler also worked out analogous laws of motion to those of Newton, see Euler's laws of motion. These extend the scope of Newton's laws to rigid bodies, but are essentially the same as above. A new equation Euler formulated is:[10]

\mathbf{I} \cdot \boldsymbol{\alpha} + \boldsymbol{\omega} \times \left ( \mathbf{I} \cdot \boldsymbol{\omega} \right ) = \boldsymbol{\tau} \,\!

where I is the moment of inertia tensor.

General planar motion

The previous equations for planar motion can be used here: corollaries of momentum, angular momentum etc. can immediately follow by applying the above definitions. For any object moving in any path in a plane,

\mathbf{r}= \bold{r}(t) = r\bold{\hat{e}}_r \,\!

the following general results apply to the particle.

Kinematics Dynamics
Position

\mathbf{r} =\bold{r}\left ( r,\theta, t \right ) = r \bold{\hat{e}}_r

Velocity
\mathbf{v} = \bold{\hat{e}}_r \frac{\mathrm{d} r}{\mathrm{d}t} + r \omega \bold{\hat{e}}_\theta
Momentum
\mathbf{p} = m \left(\bold{\hat{e}}_r \frac{\mathrm{d} r}{\mathrm{d}t} + r \omega \bold{\hat{e}}_\theta \right)

Angular momenta \mathbf{L} = m \bold{r}\times \left(\bold{\hat{e}}_r \frac{\mathrm{d} r}{\mathrm{d}t} + r \omega \bold{\hat{e}}_\theta \right)

Acceleration
\mathbf{a} =\left ( \frac{\mathrm{d}^2 r}{\mathrm{d}t^2} - r\omega^2\right )\bold{\hat{e}}_r + \left ( r \alpha + 2 \omega \frac{\mathrm{d}r}_\theta
The centripetal force is
\mathbf{F}_\bot = - m \omega^2 R \bold{\hat{e}}_r= - \omega^2 \mathbf{m} \,\!

where again m is the mass moment, and the coriolis force is

\mathbf{F}_c = 2\omega m \frac_\theta = 2\omega m v \bold{\hat{e}}_\theta \,\!

The Coriolis acceleration and force can also be written:

\mathbf{F}_c = m\mathbf{a}_c = -2 m \boldsymbol{ \omega \times v}

Central force motion

For a massive body moving in a central potential due to another object, which depends only on the radial separation between the centres of masses of the two objects, the equation of motion is:

\frac{d^2}{d\theta^2}\left(\frac{1}{\mathbf{r}}\right) + \frac{1}{\mathbf{r}} = -\frac{\mu\mathbf{r}^2}{\mathbf{l}^2}\mathbf{F}(\mathbf{r})

Equations of motion (constant acceleration)

These equations can be used only when acceleration is constant. If acceleration is not constant then the general calculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).

Linear motion Angular motion
v = v_0+at \, \omega _1 = \omega _0 + \alpha t \,
s = \frac {1} {2}(v_0+v) t \theta = \frac{1}{2}(\omega _0 + \omega _1)t
s = v_0 t + \frac {1} {2} a t^2 \theta = \omega _0 t + \frac{1}{2} \alpha t^2
v^2 = v_0^2 + 2 a s \, \omega _1^2 = \omega _0^2 + 2\alpha\theta
s = v t - \frac{1}{2} a t^2 \theta = \omega _1 t - \frac{1}{2} \alpha t^2

Galilean frame transforms

For classical (Galileo-Newtonian) mechanics, the transformation law from one inertial or accelerating (including rotation) frame (reference frame traveling at constant velocity - including zero) to another is the Galilean transform.

Unprimed quantities refer to position, velocity and acceleration in one frame F; primed quantities refer to position, velocity and acceleration in another frame F' moving at translational velocity V or angular velocity Ω relative to F. Conversely F moves at velocity (—V or —Ω) relative to F'. The situation is similar for relative accelerations.

Motion of entities Inertial frames Accelerating frames
Translation

V = Constant relative velocity between two inertial frames F and F'.
A = (Variable) relative acceleration between two accelerating frames F and F'.

Relative position
\mathbf{r}' = \mathbf{r} + \mathbf{V}t \,\!

Relative velocity
\mathbf{v}' = \mathbf{v} + \mathbf{V} \,\!
Equivalent accelerations
\mathbf{a}' = \mathbf{a}

Relative accelerations
\mathbf{a}' = \mathbf{a} + \mathbf{A}

Apparent/fictitious forces
\mathbf{F}' = \mathbf{F} - \mathbf{F}_\mathrm{app}

Rotation

Ω = Constant relative angular velocity between two frames F and F'.
Λ = (Variable) relative angular acceleration between two accelerating frames F and F'.

Relative angular position
\theta' = \theta + \Omega t \,\!

Relative velocity
\boldsymbol{\omega}' = \boldsymbol{\omega} + \boldsymbol{\Omega} \,\!
Equivalent accelerations
\boldsymbol{\alpha}' = \boldsymbol{\alpha}

Relative accelerations
\boldsymbol{\alpha}' = \boldsymbol{\alpha} + \boldsymbol{\Lambda}

Apparent/fictitious torques
\boldsymbol{\tau}' = \boldsymbol{\tau} - \boldsymbol{\tau}_\mathrm{app}

Transformation of any vector T to a rotating frame

\frac \,\!

Linear unforced DHO
  • k = spring constant
  • b = Damping coefficient
\omega' = \sqrt{\frac{k}{m}-\left ( \frac{b}{2m} \right )^2 } \,\!
Low amplitude angular SHO
  • I = Moment of inertia about oscillating axis
  • κ = torsion constant
\omega = \sqrt{\frac{\kappa}{I}}\,\!
Low amplitude simple pendulum
  • L = Length of pendulum
  • g = Gravitational acceleration
  • Θ = Angular amplitude
Approximate value

\omega = \sqrt{\frac{g}{L}}\,\!

Exact value can be shown to be:
\omega = \sqrt{\frac{g}{L}} \left [ 1 + \sum_{k=1}^\infty \frac{\prod_{n=1}^k \left ( 2n-1 \right )}{\prod_{n=1}^m \left ( 2n \right )} \sin^{2n} \Theta \right ]\,\!

Energy in mechanical oscillations
Physical situation Nomenclature Equations
SHM energy
  • T = kinetic energy
  • U = potential energy
  • E = total energy
Potential energy

U = \frac{m}{2} \left ( x \right )^2 = \frac{m \left( \omega A \right )^2}{2} \cos^2(\omega t + \phi)\,\! Maximum value at x = A:
U_\mathrm{max} \frac{m}{2} \left ( \omega A \right )^2 \,\!

Kinetic energy
T = \frac{\omega^2 m}{2} \left ( \frac{\mathrm{d} x}{\mathrm{d} t} \right )^2 = \frac{m \left ( \omega A \right )^2}{2}\sin^2\left ( \omega t + \phi \right )\,\!

Total energy
E = T + U \,\!

DHM energy E = \frac{m \left ( \omega A \right )^2}{2}e^{-bt/m} \,\!

See also

Notes

  1. ^ Mayer, Sussman & Wisdom 2001, p. xiii
  2. ^ Berkshire & Kibble 2004, p. 1
  3. ^ Berkshire & Kibble 2004, p. 2
  4. ^ Arnold 1989, p. v
  5. ^ Section: Moments and center of mass
  6. ^
  7. ^ "Relativity, J.R. Forshaw 2009"
  8. ^ "Mechanics, D. Kleppner 2010"
  9. ^ "Relativity, J.R. Forshaw 2009"
  10. ^ "Relativity, J.R. Forshaw 2009"

References

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.