constants:
g = 9.80
m
G = 6.67 × 10−11
s2
Nm2
kg 2
vectors:
!" !"
!" !"
scalar product : A ⋅ B = Ax Bx + Ay By + Az Bz = A B cosθ
!" !"
vector product : A × B = Ay Bz − Az By iˆ + Az Bx − Ax Bz ĵ + Ax By − Ay Bx k̂
general motion:
!
r = x iˆ + y ĵ + z k̂ = r r̂
kinematics:
(
) (
(
)
!
! dr
v=
= vx iˆ + v y ĵ + vz k̂
dt
!
!
! dv d 2 r
a=
=
= ax iˆ + a y ĵ + az k̂
dt dt 2
v 2f − vo2 ⎛ vo + v f ⎞
1
linear : x f − xo = vot + at 2 =
=⎜
⎟t
2
2a
⎝ 2 ⎠
v f − vo = at
ω 2f − ω o2 ⎛ ω o + ω f ⎞
1
rotational : θ f − θ o = ω ot + α t 2 =
=⎜
⎟t
2
2α
2
⎝
⎠
special motion:
force types:
torque:
Newton’s 2nd law:
impulse-momentum:
v2
v2
projectile : R = o sin ( 2θ ) circular : ac =
= rω 2
g
r
contact ( normal ) :
work-energy:
(
) (
)
!"
"
d !"
linear : F net = ma cm = p cm
dt
(
motion on a rotating body or perfect rolling : s = rθ , v = rω , a = rα
B !" !
1
1
linear : W A→B = ∫ F ⋅ dl = ΔKElin = mv B2 − mv 2A
2
2
A
dW !" !
= F ⋅v
dt
B!
!
1
1
rotational : W A→B = ∫ τ ⋅ dθ = ΔKErot = Iω B2 − Iω 2A
2
2
A
dW ! !"
= τ ⋅ω
dt
linear : KE =
potential energies:
gravity PE ( near surface of Earth ) :U ( y ) = mgy + U o
rotational inertia:
!
!
!
v A rel C = v A rel B + v B rel C
relative : !
!
v A rel B = −v B rel A
!"
!
F spring = −kΔx
tension : T
!"
!"
!"
"
"
external impulse = ∫ F net dt = Δ p system , p system = ∑ mi v i = M v cm
!
"!
"!
"! "!
!"
external impulse = ∫ τ net dt = Δ Lsystem , Lsystem = ∑ ri × p i = I ω
linear :
kinetic energy:
center of mass:
α = constant
!
!" d !"
rotational : τ net = I α = L
dt
linear : P =
conservative force:
ω f − ωo = αt
)
power:
energy conservation:
a = constant
N
kinetic / static friction : f k = µ k N
f s ≤ µs N
! ! !"
! ! !"
τ = r × F = yFz − zFy iˆ + zFx − xFz ĵ + xFy − yFx k̂ τ = r F sin θ = F⊥ r = Fr⊥
rotational :
linear/rotation link:
elastic ( spring ) :
gravity : W = mg
!" !" !" !"
A × B = A B sin θ
)
rotational : P =
1 2 p2
mv =
2
2m
rotational : KE =
external / internal model :
1 2 L2
Iω =
2
2I
total : KE =
1 2 1
1
mv + I ω 2 = I fixed ω 2
2 cm 2 cm
2 point
spring ( elastic ) PE :U ( x ) =
1
k x − xo
2
(
)2
Wext = ΔEint = ΔKE + ΔPE + ΔEthermal
mechanical energy conserved : KEbefore + PEbefore = KEafter + PEafter
!" !
W ="
∫ F ⋅ dl = 0
!"
!"
∂U ˆ ∂U
∂U
F = −∇U = −
i−
ĵ −
k̂ W A→B = −ΔU = − U B − U A
∂x
∂y
∂z
!
!
!
m1 r1 + m2 r 2 + ...
m x + m2 x2 + ...
particles or centers of objects : r cm =
⇒ xcm = 1 1
,
M
M
!
1 !
1
extended rigid objects : r cm =
r dm ⇒ xcm =
x dm, dm = λ ( x ) dx,
∫
M
M∫
(
point particles : I = m1r12 + m2r22 + ...,
additivity ( same axis ) : Itot = I1 + I 2 + ...,
)
M = m1 + m2 + ...
M = ∫ dm
continuous rigid object : I = ∫ r 2 dm,
dm = λ ( x ) dx
parallel − axis theorem : I new = I cm + Md 2
point mass, thin hoop or hollow cylinder a distance R from the axis : I = MR 2
rotational inertias:
thin rod of length L...
sphere about its center...
gravitation:
harmonic motion:
1
MR 2
2
1
about its end : I = ML2
3
2
solid : I = MR 2
5
disk or solid cylinder... about its axis : I =
!" GMm
−GMm
Newton : F = 2 ( − r̂ ) , U ( r ) =
r
r
3
MR 2
2
1
about its center : I =
ML2
12
2
hollow, thin shell : I = MR 2
3
about its edge : I =
Kepler 's 3rd law :
R3
T
2
= constant =
mass on spring : x ( t ) = Asin (ω t + φ ) ,
A = xmax , ω =
k 2π
=
m T
pendulum : θ ( t ) = Asin (ω t + φ ) ,
A = θ max , ω =
mgd 2π
=
I
T
GM
( 2π )2
Etot =
1 2
kA
2
vmax = ω A