This page gives a summary of important equations in
classical mechanics.
Nomenclature
- a = acceleration (m/s^{2})
- F = force (N = kg m/s^{2})
- KE = kinetic energy (J = kg m^{2}/s^{2})
- m = mass (kg)
- p = momentum (kg m/s)
- s = position (m)
- t = time (s)
- v = velocity (m/s)
- v_{0} = velocity at time t=0
- W = work (J = kg m^{2}/s^{2})
- s(t) = position at time t
- s_{0} = 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
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.
- <math>\mathbf{v}_{\hbox{average}} = {\Delta \mathbf{s} \over \Delta t}</math>
- <math>\mathbf{v} = {d\mathbf{s} \over dt}</math>
- a_{average} = Δv/Δt
- a = dv/dt = d^{2}s/dt^{2}
- |a_{c}| = ω^{2}R = v^{2} / R
(R = radius of the circle, ω = v/R [angular velocity])
- p = mv
- ∑F = dp/dt = d(mv)/dt
- ∑F = ma (Constant Mass)
- J = Δp = ∫Fdt
- J = FΔt if F is constant
For a single axis of rotation:
- |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
- ∑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 = d
ω/dt
- ΔKE = ∫F_{net}·ds
- KE = ∫v·dp = 1/2 mv^{2} 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.
Useful derived equations
- s(t) = 1/2at^{2} + v_{0}t + s_{0} if a is constant.
- v^{2}=v_{0}^{2} + 2a·Δs
All Wikipedia text
is available under the
terms of the GNU Free Documentation License