Classical Mechanics/Equations

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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
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

     n

sCM = (∑misi)/mtotal

    i=0

where n is the number of mass particles.

or:

sCM = (∫sρ(s)dV)/mtotal

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

Velocity

vaverage = Δs/Δt
v = ds/dt

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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