List of equations in classical mechanics

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.

Classical mechanics is the branch of physics used to describe the motion of macroscopic objects. 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. 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.

Classical mechanics utilises many equations—as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equations, manifolds, Lie groups, and ergodic theory. This page gives a summary of the most important of these.

Equations

Name of equation	Equation	Year derived	Derived by	Notes
Center of mass	Discrete case: $\mathbf{s}_{\hbox{CM}} = {1 \over m_{\hbox{total}}} \sum_{i = 1}^{n} m_i \mathbf{s}_i$ where n is the number of mass particles. 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	1687	Isaac Newton

Velocity

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

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

average velocity = change in distance / change in time.

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 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:

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

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.

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

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)

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\$

$\Delta E_p = mg\Delta h \quad\ \,\!$ in field of gravity

Central force motion

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

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.

$v = v_0+at \,$

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

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

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

These equations can be adapted for angular motion, where angular acceleration is constant:

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

$\theta = \frac{1}{2}(\omega _0 + \omega _1)t$

$\theta = \omega _0 t + \frac{1}{2} \alpha t^2$

$\omega _1^2 = \omega _0^2 + 2\alpha\theta$

$\theta = \omega _1 t - \frac{1}{2} \alpha t^2$

See also

• Acoustics
• Classical mechanics
• Electromagnetism
• Mechanics
• Optics
• Thermodynamics

References

• Arnold, Vladimir I. (1989), Mathematical Methods of Classical Mechanics (2nd ed.), Springer, ISBN 978-0-387-96890-2 
• Berkshire, Frank H.; Kibble, T. W. B. (2004), Classical Mechanics (5th ed.), Imperial College Press, ISBN 978-1860944352 
• Mayer, Meinhard E.; Sussman, Gerard J.; Wisdom, Jack (2001), Structure and Interpretation of Classical Mechanics, MIT Press, ISBN 978-0262194556 

External links

• Lectures on classical mechanics
• Biography of Isaac Newton, a key contributor to classical mechanics