Definitions

# 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)
Ek = kinetic energy (J = kg m²/s²)
Ep = potential energy (J = kg m²/s²)
m = mass (kg)
p = momentum (kg m/s)
s = displacement (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
runit = unit vector pointing from the origin in polar coordinates
θunit = 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 equation—as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equationss, 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:
$mathbf\left\{s\right\}_\left\{hbox\left\{CM\right\}\right\} = \left\{1 over m_\left\{hbox\left\{total\right\}\right\}\right\} sum_\left\{i = 0\right\}^\left\{n\right\} m_i mathbf\left\{s\right\}_i$
where n is the number of mass particles.

Continuous case:

$mathbf\left\{s\right\}_\left\{hbox\left\{CM\right\}\right\} = \left\{1 over m_\left\{hbox\left\{total\right\}\right\}\right\} int rho\left(mathbf\left\{s\right\}\right) dV$
where ρ(s) is the scalar mass density as a function of the position vector
1687 Isaac Newton

### Velocity

$mathbf\left\{v\right\}_\left\{mbox\left\{average\right\}\right\} = \left\{Delta mathbf\left\{d\right\} over Delta t\right\}$
$mathbf\left\{v\right\} = \left\{dmathbf\left\{s\right\} over dt\right\}$

### Acceleration

$mathbf\left\{a\right\}_\left\{mbox\left\{average\right\}\right\} = frac\left\{Deltamathbf\left\{v\right\}\right\}\left\{Delta t\right\}$
$mathbf\left\{a\right\} = frac\left\{dmathbf\left\{v\right\}\right\}\left\{dt\right\} = frac\left\{d^2mathbf\left\{s\right\}\right\}\left\{dt^2\right\}$

• Centripetal Acceleration

$|mathbf\left\{a\right\}_c | = omega^2 R = v^2 / R$
(R = radius of the circle, ω = v/R angular velocity)

### Momentum

$mathbf\left\{p\right\} = mmathbf\left\{v\right\}$

### Force

$sum mathbf\left\{F\right\} = frac\left\{dmathbf\left\{p\right\}\right\}\left\{dt\right\} = frac\left\{d\left(mmathbf\left\{v\right\}\right)\right\}\left\{dt\right\}$

$sum mathbf\left\{F\right\} = mmathbf\left\{a\right\} quad$   (Constant Mass)

### Impulse

$mathbf\left\{J\right\} = Delta mathbf\left\{p\right\} = int mathbf\left\{F\right\} dt$
$mathbf\left\{J\right\} = mathbf\left\{F\right\} Delta t quad$
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:

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

### Angular momentum

$|L| = mvr quad$   if v is perpendicular to r

Vector form:

$mathbf\left\{L\right\} = mathbf\left\{r\right\} times mathbf\left\{p\right\} = mathbf\left\{I\right\}, omega$

(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)

### Torque

$sum boldsymbol\left\{tau\right\} = frac\left\{dmathbf\left\{L\right\}\right\}\left\{dt\right\}$
$sum boldsymbol\left\{tau\right\} = mathbf\left\{r\right\} times mathbf\left\{F\right\} quad$
if |r| and the sine of the angle between r and p remains constant.
$sum boldsymbol\left\{tau\right\} = mathbf\left\{I\right\} boldsymbol\left\{alpha\right\}$
This one is very limited, more added later. α = dω/dt

### Precession

Omega is called the precession angular speed, and is defined:

$boldsymbol\left\{Omega\right\} = frac\left\{wr\right\}\left\{Iboldsymbol\left\{omega\right\}\right\}$

(Note: w is the weight of the spinning flywheel)

### Energy

for m as a constant:

$Delta E_k = int mathbf\left\{F\right\}_\left\{mbox\left\{net\right\}\right\} cdot dmathbf\left\{s\right\} = int mathbf\left\{v\right\} cdot dmathbf\left\{p\right\} = begin\left\{matrix\right\}frac\left\{1\right\}\left\{2\right\}end\left\{matrix\right\} mv^2 - begin\left\{matrix\right\}frac\left\{1\right\}\left\{2\right\}end\left\{matrix\right\} m\left\{v_0\right\}^2 quad$

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

### Central force motion

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

### 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.

$v = v_0+at ,$

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

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

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

$s = vt - frac \left\{1\right\} \left\{2\right\} a t^2$

Derivation of these equation in vector format and without having is shown here These equations can be adapted for angular motion, where angular acceleration is constant:

$omega _1 = omega _0 + alpha t ,$

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

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

$omega _1^2 = omega _0^2 + 2alphatheta$

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

## References

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