List Of Equations In Classical Mechanics

Nomenclature

a = acceleration (m/s²)

F = force (N = kg m/s²)

KE = kinetic energy (J = kg m²/s²)

m = mass (kg)

p = momentum (kg m/s)

s = position (m)

t = time (s)

v = velocity (m/s)

v₀ = velocity at time t=0

W = work (J = kg m²/s²)

s(t) = position at time t

s₀ = position at time t=0

r_unit = unit vector pointing from the origin in polar coordinates

θ_unit = unit vector pointing in the direction of increasing values of theta in polor coordinates

Defining Equations

Center of Mass

$\backslash mathbf\{s\}\_\{\backslash hbox\{CM\}\}\; =\; \{1\; \backslash over\; m\_\{\backslash hbox\{total\}\}\}\; \backslash sum\_\{i\; =\; 0\}^\{n\}\; m\_i\; \backslash mathbf\{s\}\_i$

$\backslash mathbf\{s\}\_\{\backslash hbox\{CM\}\}\; =\; \{1\; \backslash over\; m\_\{\backslash hbox\{total\}\}\}\; \backslash int\; \backslash rho(\backslash mathbf\{s\})\; dV$

Velocity

$\backslash mathbf\{v\}\_\{\backslash mbox\{average\}\}\; =\; \{\backslash Delta\; \backslash mathbf\{s\}\; \backslash over\; \backslash Delta\; t\}$

$\backslash mathbf\{v\}\; =\; \{d\backslash mathbf\{s\}\; \backslash over\; dt\}$

Acceleration

$\backslash mathbf\{a\}\_\{\backslash mbox\{average\}\}\; =\; \backslash frac\{\backslash Delta\backslash mathbf\{v\}\}\{\backslash Delta\; t\}$

$\backslash mathbf\{a\}\; =\; \backslash frac\{d\backslash mathbf\{v\}\}\{dt\}\; =\; \backslash frac\{d^2\backslash mathbf\{s\}\}\{dt^2\}$

• Centripetal Acceleration

$|\backslash mathbf\{a\}\_c\; |\; =\; \backslash omega^2\; R\; =\; v^2\; /\; R$

Momentum

$\backslash mathbf\{p\}\; =\; m\backslash mathbf\{v\}$

Force

$\backslash sum\; \backslash mathbf\{F\}\; =\; \backslash frac\{d\backslash mathbf\{p\}\}\{dt\}\; =\; \backslash frac\{d(m\backslash mathbf\{v\})\}\{dt\}$

$\backslash sum\; \backslash mathbf\{F\}\; =\; m\backslash mathbf\{a\}\; \backslash quad\backslash$   (Constant Mass)

Impulse

$\backslash mathbf\{J\}\; =\; \backslash Delta\; \backslash mathbf\{p\}\; =\; \backslash int\; \backslash mathbf\{F\}\; dt$

$\backslash mathbf\{J\}\; =\; \backslash mathbf\{F\}\; \backslash Delta\; t\; \backslash quad\backslash$

  if F is constant

Moment of Intertia

Angular Momentum

$|L|\; =\; mvr\; \backslash quad\backslash$   if v is perpendicular to r

$\backslash mathbf\{L\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{p\}\; =\; \backslash mathbf\{I\}\backslash ,\; \backslash omega$

Torque

$\backslash sum\; \backslash boldsymbol\{\backslash tau\}\; =\; \backslash frac\{d\backslash mathbf\{L\}\}\{dt\}$

$\backslash sum\; \backslash boldsymbol\{\backslash tau\}\; =\; \backslash mathbf\{r\}\; \backslash times\; \backslash mathbf\{F\}\; \backslash quad$

$\backslash sum\; \backslash boldsymbol\{\backslash tau\}\; =\; \backslash mathbf\{I\}\; \backslash boldsymbol\{\backslash alpha\}$

Precession

Energy

$\backslash Delta\; \backslash mbox\{\; KE\; \}\; =\; \backslash int\; \backslash mathbf\{F\}\_\{\backslash mbox\{net\}\}\; \backslash cdot\; d\backslash mathbf\{s\}$

$\backslash mbox\{KE\; \}\; =\; \backslash int\; \backslash mathbf\{v\}\; \backslash cdot\; d\backslash mathbf\{p\}\; =\; \backslash begin\{matrix\}\backslash frac\{1\}\{2\}\backslash end\{matrix\}\; mv^2\; \backslash quad\backslash$   if m is constant

$\backslash mbox\{PE\}\_\{\backslash mbox\{due\; to\; gravity\}\}\; =\; mgh\; \backslash quad\backslash$

  (near the earth's surface)

Central Force Motion

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

Gravitational Force

$\backslash mathbf\{F(r)\}\; =\; -\backslash frac\{\backslash mathbf\{Gm\_1\}\backslash mathbf\{m\_2\}\}\{\backslash mathbf\{r^2\}\}$

G is the gravitational constant, one of the physical constants

Useful derived equations

Position of an accelerating body

$\backslash mathbf\{s\}(t)\; =\; \backslash begin\{matrix\}\backslash frac\{1\}\{2\}\backslash end\{matrix\}\; \backslash mathbf\{a\}\; t^2\; +\; \backslash mathbf\{v\}\_0\; t\; +\; \backslash mathbf\{s\}\_0\; \backslash quad\backslash$   if a is constant.

Equation for velocity

$v^2\; =v\_0^2\; +\; 2\backslash mathbf\{a\}\; \backslash cdot\; \backslash Delta\backslash mathbf\{s\}$

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