List of equations in classical mechanics

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

1 Nomenclature
2 Defining equations
3 Useful derived equations
- 3.1 Equations of motion (constant acceleration)

[edit] Nomenclature

a = acceleration (m/s²)

g = gravitational field strength/acceleration in free-fall (m/s²)

F = force (N = kg m/s²)

E_k = kinetic energy (J = kg m²/s²)

E_p = potential energy (J = kg m²/s²)

m = mass (kg)

p = momentum (kg m/s)

s = displacement (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 (m N, not J) (torque is the rotational form of force)

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 polar coordinates

Note: All quantities in bold represent vectors.

[edit] Defining equations

[edit] Center of mass

In the discrete case:

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

where $n$ is the number of mass particles.

Or in the continuous case:

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

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

[edit] Velocity

$\mathbf{v}_{\mbox{average}} = {\Delta \mathbf{d} \over \Delta t}$

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

[edit] Acceleration

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

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

Centripetal Acceleration

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

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

[edit] Momentum

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

[edit] Force

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

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

[edit] Impulse

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

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

if F is constant

[edit] 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:

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

[edit] Angular momentum

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

Vector form:

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

(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.

[edit] Torque

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

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

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

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

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

[edit] Precession

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

$\boldsymbol{\Omega} = \frac{wr}{I\boldsymbol{\omega}}$

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

[edit] Energy

for m as a constant:

$\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\$