Classical Mechanics/Equations

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

---

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
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 polor coordinates

Note: All quantities in bold represent vectors.

Defining Equations

Center of Mass

      n

s_CM = (∑m_is_i)/m_total

     i=0

where n is the number of mass particles.

or:

s_CM = (∫sρ(s)dV)/m_total

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

Velocity

v_average = Δs/Δt
v = ds/dt

Acceleration

a_average = Δv/Δt
a = dv/dt = d²s/dt²

• Centripetal Acceleration

|a_c| = 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 = ∫F_net·ds

KE = ∫v·dp = 1/2 mv² if m is constant

PE_{due 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

/Talk