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List of equations in classical mechanics

This page gives a summary of important equations in classical mechanics.

Table of contents

Nomenclature

a = acceleration (m/s2)
F = force (N = kg m/s2)
KE = kinetic 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)
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:

<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.

Velocity

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

Acceleration

aaverage = Δv/Δt
a = dv/dt = d2s/dt2

  • Centripetal Acceleration

|ac| = ω2R = v2 / R
(R = radius of the circle, ω = v/R [angular velocity])

Momentum

p = mv

Force

F = dp/dt = d(mv)/dt

F = ma (Constant Mass)

Impulse

J = Δp = ∫Fdt
J = FΔt if F is constant

Moment of Intertia

For a single axis of rotation:

Angular Momentum

|L| = mvr iff v is perpendicular to r

Vector form:

L = r×p = Iω

(Note: I can be treated like a vector if it is diagonalized first, but it is actually a 3×3 matrix)

r is the radius vector

Torque

τ = dL/dt
τ = r×F
if |r| and the sine of the angle between r and p remains constant.
τ = Iα
This one is very limited, more added later. α = dω/dt

Precession

Energy

ΔKE = ∫Fnet·ds

KE = ∫v·dp = 1/2 mv2 if m is constant

PEdue to gravity = mgh (near the earth's surface)

g is the acceleration due to gravity, one the physical constants.

Central Force Motion

Useful derived equations

Position of an accelerating body

s(t) = 1/2at2 + v0t + s0 if a is constant.

Equation for velocity

v2=v02 + 2a·Δs



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