List of equations in classical mechanics

This page gives a summary of important 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.

Defining equations

Center of mass

In the discrete case:

$\mathbf {s} _{\hbox{CM}}={1 \over m_{\hbox{total}}}\sum _{i=0}^{n}m_{i}\mathbf {s} _{i}$

where $n$ is the number of mass particles.

Or in the 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

Velocity

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

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

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

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

Useful derived equations

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.

$v=v_{0}+at\,$

$s={\frac {1}{2}}(v_{0}+v)t$

$s=v_{0}t+{\frac {1}{2}}at^{2}$

$v^{2}=v_{0}^{2}+2as\,$

$s=vt-{\frac {1}{2}}at^{2}$

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

$\omega _{1}=\omega _{0}+\alpha t\,$

$\theta ={\frac {1}{2}}(\omega _{0}+\omega _{1})t$

$\theta =\omega _{0}t+{\frac {1}{2}}\alpha t^{2}$

$\omega _{1}^{2}=\omega _{0}^{2}+2\alpha \theta$

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