# List of equations in classical mechanics

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List of equations in classical mechanics

;Nomenclature

: "a" = acceleration (m/s&sup2;): "g" = gravitational field strength/acceleration in free-fall (m/s&sup2;): "F" = force (N = kg m/s&sup2;): "E"k = kinetic energy (J = kg m&sup2;/s&sup2;): "E"p = potential energy (J = kg m&sup2;/s&sup2;): "m" = mass (kg): "p" = momentum (kg m/s): "s" = displacement (m): "R" = radius (m): "t" = time (s): "v" = velocity (m/s): "v"0 = velocity at time t=0: "W" = work (J = kg m&sup2;/s&sup2;): "&tau;" = torque (m N, not J) (torque is the rotational form of force): "s"(t) = position at time t: "s"0 = position at time t=0: "r"unit = unit vector pointing from the origin in polar coordinates: "&theta;"unit = unit vector pointing in the direction of increasing values of theta in polar coordinates

Note: All quantities in bold represent vectors.

Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. [Harvnb|Mayer|Sussman|Wisdom|2001|p=xiii] 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. [Harvnb|Berkshire|Kibble|2004|p=1] 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. [Harvnb|Berkshire|Kibble|2004|p=2]

Classical mechanics utilises many equation&mdash;as well as other mathematical concepts&mdash;which relate various physical quantities to one another. These include differential equationss, manifolds, Lie groups, and ergodic theory. [Harvnb|Arnold|1989|p=v] This page gives a summary of the most important of these.

Equations

Velocity

: $mathbf\left\{v\right\}_\left\{mbox\left\{average = \left\{Delta mathbf\left\{d\right\} over Delta t\right\}$: $mathbf\left\{v\right\} = \left\{dmathbf\left\{s\right\} over dt\right\}$

Acceleration

: $mathbf\left\{a\right\}_\left\{mbox\left\{average = frac\left\{Deltamathbf\left\{v\left\{Delta t\right\}$ : $mathbf\left\{a\right\} = frac\left\{dmathbf\left\{v\left\{dt\right\} = frac\left\{d^2mathbf\left\{s\left\{dt^2\right\}$

*Centripetal Acceleration

: $|mathbf\left\{a\right\}_c | = omega^2 R = v^2 / R$("R" = radius of the circle, ω = "v/R" angular velocity)

Momentum

: $mathbf\left\{p\right\} = mmathbf\left\{v\right\}$

Force

:$sum mathbf\left\{F\right\} = frac\left\{dmathbf\left\{p\left\{dt\right\} = frac\left\{d\left(mmathbf\left\{v\right\}\right)\right\}\left\{dt\right\}$

:$sum mathbf\left\{F\right\} = mmathbf\left\{a\right\} quad$ (Constant Mass)

Impulse

:$mathbf\left\{J\right\} = Delta mathbf\left\{p\right\} = int mathbf\left\{F\right\} dt$:

$mathbf\left\{J\right\} = mathbf\left\{F\right\} Delta t quad$
if F is constant

Moment of inertia

For a single axis of rotation:The moment of inertia for an object is the sum of the products of the mass element and the square of their distances from the axis of rotation:

$I = sum r_i^2 m_i =int_M r^2 mathrm\left\{d\right\} m = iiint_V r^2 ho\left(x,y,z\right) mathrm\left\{d\right\} V$

Angular momentum

:$|L| = mvr quad$ if v is perpendicular to r

Vector form:

:$mathbf\left\{L\right\} = mathbf\left\{r\right\} imes mathbf\left\{p\right\} = mathbf\left\{I\right\}, omega$

(Note: I can be treated like a vector if it is diagonalized first, but it is actually a 3&times;3 matrix - a tensor of rank-2)

Torque

::if |r| and the sine of the angle between r and p remains constant.:This one is very limited, more added later. α = dω/dt

Precession

Omega is called the precession angular speed, and is defined:

:

(Note: w is the weight of the spinning flywheel)

Energy

for "m" as a constant:

:

:$Delta E_p = mgDelta h quad ,!$ in field of gravity

Central force motion

: $frac\left\{d^2\right\}\left\{d heta^2\right\}left\left(frac\left\{1\right\}\left\{mathbf\left\{r ight\right) + frac\left\{1\right\}\left\{mathbf\left\{r = -frac\left\{mumathbf\left\{r\right\}^2\right\}\left\{mathbf\left\{l\right\}^2\right\}mathbf\left\{F\right\}\left(mathbf\left\{r\right\}\right)$

Equations of motion (constant acceleration)

These equations can be used only when acceleration is constant. If acceleration is not constant then calculus must be used.

:$v = v_0+at ,$

:$s = frac \left\{1\right\} \left\{2\right\}\left(v_0+v\right) t$

:$s = v_0 t + frac \left\{1\right\} \left\{2\right\} a t^2$

:$v^2 = v_0^2 + 2 a s ,$

:$s = vt - frac \left\{1\right\} \left\{2\right\} a t^2$

Derivation of these equation in vector format and without having is shown here These equations can be adapted for angular motion, where angular acceleration is constant:

: $omega _1 = omega _0 + alpha t ,$

: $heta = frac\left\{1\right\}\left\{2\right\}\left(omega _0 + omega _1\right)t$

: $heta = omega _0 t + frac\left\{1\right\}\left\{2\right\} alpha t^2$

: $omega _1^2 = omega _0^2 + 2alpha heta$

: $heta = omega _1 t - frac\left\{1\right\}\left\{2\right\} alpha t^2$

ee also

*Acoustics
*Classical mechanics
*Electromagnetism
*Mechanics
*Optics
*Thermodynamics

Notes

References

*citation|title=Mathematical Methods of Classical Mechanics|last=Arnold|first=Vladimir I.|publisher=Springer|year=1989|isbn=978-0-387-96890-2|edition=2nd
*citation|title=Classical Mechanics|last1=Berkshire|last2=Kibble|first1=Frank H.|first2=T. W. B.|edition=5th|publisher=Imperial College Press|year=2004|isbn=978-1860944352
*citation|title=Structure and Interpretation of Classical Mechanics|last1=Mayer|last2=Sussman|last3=Wisdom|first1=Meinhard E.|first2=Gerard J.|first3=Jack|publisher=MIT Press|year=2001|isbn=978-0262194556

* [http://www.astro.uvic.ca/~tatum/classmechs.html Lectures on classical mechanics]
* [http://scienceworld.wolfram.com/biography/Newton.html Biography of Isaac Newton, a key contributor to classical mechanics]

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