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 Position of an accelerating body
- 3.2 Equation for velocity

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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.

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Defining Equations

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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 vector

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Velocity

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

<math>\mathbf{v} = {d\mathbf{s} \over dt}</math>

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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)

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Momentum

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

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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)

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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

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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:

<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>

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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.

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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

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Precession

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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

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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>

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