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List of equations in classical mechanics

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

Contents

1 Nomenclature

2 Defining Equations

2.1 Center of Mass
2.2 Velocity
2.3 Acceleration
2.4 Momentum
2.5 Force
2.6 Impulse
2.7 Moment of Intertia
2.8 Angular Momentum
2.9 Torque
2.10 Precession
2.11 Energy
2.12 Central Force Motion

3 Useful derived equations

3.1 Position of an accelerating body
3.2 Equation for velocity

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

Note: All quantities in bold represent vectors.

Defining Equations

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.

Velocity

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

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

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)

Momentum

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

Force

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

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

Impulse

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

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

if F is constant

Moment of Intertia

For a single axis of rotation:

Angular Momentum

$|L| = mvr \quad\$ iff 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)

r is the radius vector

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

Precession

Energy

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

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

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

(near the earth's surface)

g is the acceleration due to gravity, one the physical constants.

Central Force Motion

Useful derived equations

Position of an accelerating body

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

Equation for velocity

$v^2 =v_0^2 + 2\mathbf{a} \cdot \Delta\mathbf{s}$

Categories: Classical mechanics

Last updated: 01-22-2005 01:27:38

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