# List of equations in classical mechanics

 Contents

## Nomenclature

a = acceleration (m/s²)
g = gravitational constant (m/s²)
F = force (N = kg m/s²)
Ek = kinetic energy (J = kg m²/s²)
Ep = potential 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)
τ = torque (J = N m) (torque is the rotational form of force)
s(t) = position at time t
s0 = position at time t=0
runit = unit vector pointing from the origin in polar coordinates
θunit = 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:

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

where [itex]n[itex] is the number of mass particles.

Or in the continuous case:

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

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

### Velocity

[itex]\mathbf{v}_{\mbox{average}} = {\Delta \mathbf{s} \over \Delta t}[itex]
[itex]\mathbf{v} = {d\mathbf{s} \over dt}[itex]

### Acceleration

[itex]\mathbf{a}_{\mbox{average}} = \frac{\Delta\mathbf{v}}{\Delta t} [itex]
[itex]\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{s}}{dt^2} [itex]
• Centripetal Acceleration
[itex] |\mathbf{a}_c | = \omega^2 R = v^2 / R [itex]

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

### Momentum

[itex]\mathbf{p} = m\mathbf{v}[itex]

### Force

[itex] \sum \mathbf{F} = \frac{d\mathbf{p}}{dt} = \frac{d(m\mathbf{v})}{dt} [itex]
[itex] \sum \mathbf{F} = m\mathbf{a} \quad\ [itex]   (Constant Mass)

### Impulse

[itex] \mathbf{J} = \Delta \mathbf{p} = \int \mathbf{F} dt [itex]
[itex] \mathbf{J} = \mathbf{F} \Delta t \quad\ [itex]
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:

[itex]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[itex]

### Angular Momentum

[itex] |L| = mvr \quad\ [itex]   if v is perpendicular to r

Vector form:

[itex] \mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{I}\, \omega [itex]

(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)

### Torque

[itex] \sum \boldsymbol{\tau} = \frac{d\mathbf{L}}{dt} [itex]
[itex] \sum \boldsymbol{\tau} = \mathbf{r} \times \mathbf{F} \quad [itex]

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

[itex] \sum \boldsymbol{\tau} = \mathbf{I} \boldsymbol{\alpha} [itex]

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

### Energy

m is here constant.

[itex] \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\ [itex]
[itex] \Delta E_p = mgh \quad\ \,\![itex] in field of gravity

### Central Force Motion

[itex]\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})[itex]

### Gravitational Force

[itex]\mathbf{F(r)} = -\frac{\mathbf{Gm_1}\mathbf{m_2}}{\mathbf{r^2}}[itex]
G is the gravitational constant, one of the physical constants

## Useful derived equations

### Position of an accelerating body

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

### Equation for velocity

[itex] v^2 =v_0^2 + 2\mathbf{a} \cdot \Delta\mathbf{s}[itex]

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