List of equations in classical mechanics

From Academic Kids

This page gives a summary of important equations in classical mechanics.

Contents

Nomenclature

a = acceleration (m/s²)
g = gravitational constant (m/s²)
F = force (N = kg m/s²)
Ek = kinetic energy (J = kg m²/s²)
Ep = potential energy (J = kg m²/s²)
m = mass (kg)
p = momentum (kg m/s)
s = position (m)
R = radius (m)
t = time (s)
v = velocity (m/s)
v0 = velocity at time t=0
W = work (J = kg m²/s²)
τ = torque (J = N m) (torque is the rotational form of force)
s(t) = position at time t
s0 = position at time t=0
runit = unit vector pointing from the origin in polar coordinates
θunit = unit vector pointing in the direction of increasing values of theta in polor coordinates

Note: All quantities in bold represent vectors.

Defining Equations

Center of Mass

In the discrete case:

<math>\mathbf{s}_{\hbox{CM}} = {1 \over m_{\hbox{total}}} \sum_{i = 0}^{n} m_i \mathbf{s}_i<math>

where <math>n<math> is the number of mass particles.

Or in the continuous case:

<math>\mathbf{s}_{\hbox{CM}} = {1 \over m_{\hbox{total}}} \int \rho(\mathbf{s}) dV<math>

where ρ(s) is the scalar mass density as a function of the position vecto

Velocity

<math>\mathbf{v}_{\mbox{average}} = {\Delta \mathbf{s} \over \Delta t}<math>
<math>\mathbf{v} = {d\mathbf{s} \over dt}<math>

Acceleration

<math>\mathbf{a}_{\mbox{average}} = \frac{\Delta\mathbf{v}}{\Delta t} <math>
<math>\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{s}}{dt^2} <math>
  • Centripetal Acceleration
<math> |\mathbf{a}_c | = \omega^2 R = v^2 / R <math>

(R = radius of the circle, ω = v/R angular velocity)

Momentum

<math>\mathbf{p} = m\mathbf{v}<math>

Force

<math> \sum \mathbf{F} = \frac{d\mathbf{p}}{dt} = \frac{d(m\mathbf{v})}{dt} <math>
<math> \sum \mathbf{F} = m\mathbf{a} \quad\ <math>   (Constant Mass)

Impulse

<math> \mathbf{J} = \Delta \mathbf{p} = \int \mathbf{F} dt <math>
<math> \mathbf{J} = \mathbf{F} \Delta t \quad\ <math>
  if F is constant

Moment of Intertia

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:

<math>I = \sum r_i^2 m_i =\int_M r^2 \mathrm{d} m = \iiint_V r^2 \rho(x,y,z) \mathrm{d} V<math>

Angular Momentum

<math> |L| = mvr \quad\ <math>   if v is perpendicular to r

Vector form:

<math> \mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{I}\, \omega <math>

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

r is the radius vector

Torque

<math> \sum \boldsymbol{\tau} = \frac{d\mathbf{L}}{dt} <math>
<math> \sum \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} \quad <math>

if |r| and the sine of the angle between r and p remains constant.

<math> \sum \boldsymbol{\tau} = \mathbf{I} \boldsymbol{\alpha} <math>

This one is very limited, more added later. α = dω/dt

Precession

Energy

m is here constant.

<math> \Delta E_k = \int \mathbf{F}_{\mbox{net}} \cdot d\mathbf{s} = \int \mathbf{v} \cdot d\mathbf{p} = \begin{matrix}\frac{1}{2}\end{matrix} mv^2 - \begin{matrix}\frac{1}{2}\end{matrix} m{v_0}^2 \quad\ <math>
<math> \Delta E_p = mgh \quad\ \,\!<math> in field of gravity

Central Force Motion

<math>\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})<math>

Gravitational Force

<math>\mathbf{F(r)} = -\frac{\mathbf{Gm_1}\mathbf{m_2}}{\mathbf{r^2}}<math>
G is the gravitational constant, one of the physical constants

Useful derived equations

Position of an accelerating body

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

Equation for velocity

<math> v^2 =v_0^2 + 2\mathbf{a} \cdot \Delta\mathbf{s}<math>
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