Vector Rules
A . B = A x B x + A y B y + A z B z
A x B = (A y B z - A z - B y ) î - (A x B z - A z - B x ) ĵ
A x B = (A y B z - A z - B y ) î - (A x B z - A z - B x ) 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 x = v x0 + a x t
x 0 = x 0 + v x0 t + 1/2 a x0 t 2
v 2 x = v 2 x0 + 2 a x t (x- x 0 )
v x = v x0 + a x t
x 0 = x 0 + v x0 t + 1/2 a x0 t 2
v 2 x = v 2 x0 + 2 a x t (x- x 0 )
if a non-constant:
v (t) = v 0 + ∫ a(t') dt'
x (t) = x 0 + ∫ 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.
x (t) = x 0 + ∫ 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 / 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.
Kinetic - m v 2 / 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
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
approximate acceleration due to gravity on earth
G = 6.7 x 10^-11 N m^2/kg^2
Gravitational constant
F = -(G M 1 M 2 )/R 12
U = -(G M 1 M 2 )/R 2 12
U = -(G M 1 M 2 )/R 2 12
Frictional Force
F fric = μ F Norm
F Norm is the normal force, which is perpendicular to the direction of movement.
F 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
W = ∫ F . ds
P = dE/dt = F . v
J = ∫ F dt = Δ p
Center of mass
R cm = Σ m i R i / Σ m i
I = Σ m i R 2 i
I = Σ m i R 2 i
Rotational Motion
if α is constant
ω = ω 0 + α t
θ = θ 0 + ω t + 1/2 α t 2
ω 2 = ω 2 0 + 2 α (θ - θ 0 )
ω = ω 0 + α t
θ = θ 0 + ω t + 1/2 α t 2
ω 2 = ω 2 0 + 2 α (θ - θ 0 )
Combined Rotational and Non-Rotational kinetic energy
K = 1/2 I ω 2 + 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
L = r x p
τ = dL/dt
τ = I α
L = I ω
W = ∫ τ d θ
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