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List of equations in classical mechanics

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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)
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)
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
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.

Defining equations

In the discrete case:

where is the number of mass particles.

Or in the continuous case:

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

Velocity

Acceleration

  • Centripetal Acceleration

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

Force

  (Constant Mass)

Impulse

  if F is constant

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:

  if v is perpendicular to r

Vector form:

(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.

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

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

Precession

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

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

Energy

for m as a constant:

in field of gravity

Central force motion

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.

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