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 2.13 Gravitational Force 3 Useful derived equations 3.1 Position of an accelerating body 3.2 Equation for velocity

Nomenclature

a = acceleration (m/s²)

g = gravitational constant (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 = 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

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:

<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>