Wednesday, July 25, 2012

Physics-Intro to Mechanics



Vector Rules
A . B = A + A + A z
A x B = (A B z - A - B ) î - (A B z - A - B ) ĵ
More information on vectors available in the Introduction to Vectors and Calculus page as well as the Introduction to Linear Algebra page.

Positional equations
if a x is a constant acceleration then the movement for some object is governed by:
= v x0 + a t
= x + v x0 t + 1/2 a x0 
= v x0 + 2 a t (x- x )
if a non-constant:
v (t) = v + ∫ a(t') dt'
x (t) = x + ∫ v(t') dt'
This is basically the same result as before, we are just integrating over a time dependent acceleration and velocity, so we take the areas under these curves to find the velocity.

Momentum and Energy
p = m v
Kinetic - m v / 2
Potential due to gravity on earth - m g h
The most important aspect of this is that the total momentum in a closed system or frame is conserved. This is the famous conservation of momentum law, or newton's second law in its more complete form.
Force
F = m a
F = dp/dt = m dv/dt + v dm/dt
Gravitation
g = 9.8 m/s^2
approximate acceleration due to gravity on earth
G = 6.7 x 10^-11 N m^2/kg^2
Gravitational constant
F = -(G M )/R 12
U = -(G M )/R 2 12
Frictional Force
fric = μ F Norm
Norm is the normal force, which is perpendicular to the direction of movement.
Parameter Relations
F = - dU/dx
W = ∫ F . ds
P = dE/dt = F . v
J = ∫ F dt = Δ p
Center of mass
cm = Σ m i R i / Σ m i
I = Σ m i
Rotational Motion
if α is constant
ω = ω + α t
θ = θ + ω t + 1/2 α t 
ω = ω + 2 α (θ - θ )
Combined Rotational and Non-Rotational kinetic energy
K = 1/2 I ω + 1/2 m v 2
τ = r x F
L = r x p
τ = dL/dt
τ = I α
L = I ω
W = ∫ τ d θ

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