Final Crib Sheet ·
Ch20 : 3D kinematics Mass Moment of Inertia General
-
w = W
,
+ Wa a = b + c -
2bc -
CoSA
w = W + Wi + dXW x
b = (32 4ac - YX
-
-
- =
,
direction 2a
change
j
& W
,
that doesn't M
k =
* A and B on same body
Pre-Midterm 1 content
-
Va =
Vi + wX Ta /
Rectilinear Motion
-
an =
Ap + XX Ta + wx(wXTAB)
variable a constant a
Wab. Nar O Of Was Tab
=
O
-
=
a = dV/d + v =
Vo + at
* A and B NOT on same
body v = dS/d S =
So + Vot + Eat
=>
(alxyz (B) xyz
=
and ads = v dv v =
vj2 + 294S
path
"
~
-
Va =
Vi + 1x
Tab +
(Xa , )xyz n, t, b coordinates
n t
-
An =
Ap + it X
Nasi + c X (dX Nasal
8
at = j
=
[1 +
(dy(ax)2y3
p
an =
V2 1 day/dx2
+ 2mx(Va / B(xyz + (aa , B)xyz
& * Anis ALWAYS pointed
towards center
* only use
rotating frame equations if
ap = 0
of curvature and and at
translation relative to a
rotating frame
r, G, z coordinates
is given , otherwise use regular equations ow[rad]
Vr = j ar = r - rin
Ch 21 :
3D Kinematics Vo = r r + 25
.
af =
m z
*
sign matters ! Vz =
=
↑2
CM equations :
az =
EmX Emy
12 * don't forget abt chain
-
X(m =
i Yom =
rule of o when
M Mrotal finding r and j
+ otal
Pulley motion analysis
Moment of Inertia : S
If n cords are involved n + / , variables must
-
2xx =
Smrx dm =
Sm(y" + z)dm be introduced
Set S as non-constant lengths
-
Eye =
SmTgdm Sm(x" + =
zam ,
-
Eze
=
Smrzdm Su(x2 = + y')dm FBD Of box on cart
Ch 21 Conta . b
.
# it car r is
Product of Inertia :
C NOT free
Exy =
Saxydm [xy
=
MXY Principal of Impulse and Momentum :
↑ rolling
2xz =
SpXzdm Xandy coordinates o r -M(Vo) ,
+
Ear =
Med Wb W + Wa
Syzdm ↓
+ F+
*
sign matters !
Zyz ↓
= 2
-
(Hol
[SModr CHo
+
P
-
= & >
-
Parallel Axis Theorem :
↑
↑
+
-
[xx =
(2x x)0 +
m(y5 +
z5) -
tomentum for static object
iso S Fys No
Ny
if A and B are two fixed points body
=5)
on
m(x)
same
-
Tyy
=
(2jy)o + +
about direction of friction,
/
# if unsure
-
angular momentum about axis AB is conserved
- -
z
=
(2 = =)o +
m(x) +
yj) -
can mix impulse + momentum M(Vol ,
= (Hola write IF equations to see if it makes sense
is
** FRICTION OPPOSES MOTION
-
If (HoA) =
(Horlz w about OA
Exy ([x y )y +
mxy
- ,
= . ,
Moment is transferrable along same
Work [J] or
[N m] .
body
Axis of symmetry same SFcosods
:
as
long as direction is variable force : U = * +Ve if F
-
If symmetric about Xz-plane = Exz # O and
↳ recalculate Zoa constant force :
U =
FlosOLS
s are
2 symmetric planes => Ixy =
Ixz =
lyz = 0 in same
Kineric Energy : weight :
U =
Way
(a) direction
Arbitrary Axis :
uz
-
Fixed point O : T =
EW :
Ho spring : u =
Ek(S2-si)
20a = [xx4x2 +
zyy4y + 2
=
↳ S, and S represents stretched/compressed
-
CMG :
T =
EMV3 +
EIxWx + IgWg +
YEzw ?
2ExyMely 2[xzUxUz 2IyzUyUz
=
-
moment : U M(
-
mod
-
=
,
T + [U . z
=
Tz (for one body)
:
eX
z Kinetic is Sayar (does Not have
to
Conservation of Energy = If NO collision
Z
Y
-
energy
origin)
T
be about
-
Do NOT use for
impact/collision Qs
T + V = y + Vav =
Vg + Ve
*
Set Datum and if :
y is o re
a = + Wy
y is below E-Wy
u = 0i + CoS30] + Sin30
corner permanently
hooks onto point
[kg m
a
Angular momentum (wrt 0 or 6) : .
1)(Hol , and Hot
-
Hx =
FxxWx -
[xyGy -
[xzWz 2) (HoA) ,
= (Ho) , Lot
3) Zon = ?
Hy FyxWx +
IyyWy [yzWz (Not principle axis)
-
Rotation
=
at
- -
:
4) [Ho1) , =
(Hot) = (Ho)2 Not ·
is first time z
-
Hz =
FzxWx -
1zywy + [zzwz * always find wand to save
-
Hy =
[xxwx -
xywy =
2xzw Z [Mx =
[xxx -
(Iy [z)wywz
- -
Exy(xy
-
wzWx)
-
[yz(wy2 (z) - =
2zx(xz +
0xwe)
Angular Moment Curt to point A) :
EMy =
zyxy
-
(2z -
[x)GzWx -
zyz(xz -
wxwy) -
[zx(wz -
wx) -
Exy(xz + Wywz)
Ma =
roXMVj + 40
[Mz =
zzXz -
(2x -
[y)wxwy -
2zx(Xx -
WyWz) -
2xy(wx wj) -
-
2yz(xy + wzwx)
·
Ch20 : 3D kinematics Mass Moment of Inertia General
-
w = W
,
+ Wa a = b + c -
2bc -
CoSA
w = W + Wi + dXW x
b = (32 4ac - YX
-
-
- =
,
direction 2a
change
j
& W
,
that doesn't M
k =
* A and B on same body
Pre-Midterm 1 content
-
Va =
Vi + wX Ta /
Rectilinear Motion
-
an =
Ap + XX Ta + wx(wXTAB)
variable a constant a
Wab. Nar O Of Was Tab
=
O
-
=
a = dV/d + v =
Vo + at
* A and B NOT on same
body v = dS/d S =
So + Vot + Eat
=>
(alxyz (B) xyz
=
and ads = v dv v =
vj2 + 294S
path
"
~
-
Va =
Vi + 1x
Tab +
(Xa , )xyz n, t, b coordinates
n t
-
An =
Ap + it X
Nasi + c X (dX Nasal
8
at = j
=
[1 +
(dy(ax)2y3
p
an =
V2 1 day/dx2
+ 2mx(Va / B(xyz + (aa , B)xyz
& * Anis ALWAYS pointed
towards center
* only use
rotating frame equations if
ap = 0
of curvature and and at
translation relative to a
rotating frame
r, G, z coordinates
is given , otherwise use regular equations ow[rad]
Vr = j ar = r - rin
Ch 21 :
3D Kinematics Vo = r r + 25
.
af =
m z
*
sign matters ! Vz =
=
↑2
CM equations :
az =
EmX Emy
12 * don't forget abt chain
-
X(m =
i Yom =
rule of o when
M Mrotal finding r and j
+ otal
Pulley motion analysis
Moment of Inertia : S
If n cords are involved n + / , variables must
-
2xx =
Smrx dm =
Sm(y" + z)dm be introduced
Set S as non-constant lengths
-
Eye =
SmTgdm Sm(x" + =
zam ,
-
Eze
=
Smrzdm Su(x2 = + y')dm FBD Of box on cart
Ch 21 Conta . b
.
# it car r is
Product of Inertia :
C NOT free
Exy =
Saxydm [xy
=
MXY Principal of Impulse and Momentum :
↑ rolling
2xz =
SpXzdm Xandy coordinates o r -M(Vo) ,
+
Ear =
Med Wb W + Wa
Syzdm ↓
+ F+
*
sign matters !
Zyz ↓
= 2
-
(Hol
[SModr CHo
+
P
-
= & >
-
Parallel Axis Theorem :
↑
↑
+
-
[xx =
(2x x)0 +
m(y5 +
z5) -
tomentum for static object
iso S Fys No
Ny
if A and B are two fixed points body
=5)
on
m(x)
same
-
Tyy
=
(2jy)o + +
about direction of friction,
/
# if unsure
-
angular momentum about axis AB is conserved
- -
z
=
(2 = =)o +
m(x) +
yj) -
can mix impulse + momentum M(Vol ,
= (Hola write IF equations to see if it makes sense
is
** FRICTION OPPOSES MOTION
-
If (HoA) =
(Horlz w about OA
Exy ([x y )y +
mxy
- ,
= . ,
Moment is transferrable along same
Work [J] or
[N m] .
body
Axis of symmetry same SFcosods
:
as
long as direction is variable force : U = * +Ve if F
-
If symmetric about Xz-plane = Exz # O and
↳ recalculate Zoa constant force :
U =
FlosOLS
s are
2 symmetric planes => Ixy =
Ixz =
lyz = 0 in same
Kineric Energy : weight :
U =
Way
(a) direction
Arbitrary Axis :
uz
-
Fixed point O : T =
EW :
Ho spring : u =
Ek(S2-si)
20a = [xx4x2 +
zyy4y + 2
=
↳ S, and S represents stretched/compressed
-
CMG :
T =
EMV3 +
EIxWx + IgWg +
YEzw ?
2ExyMely 2[xzUxUz 2IyzUyUz
=
-
moment : U M(
-
mod
-
=
,
T + [U . z
=
Tz (for one body)
:
eX
z Kinetic is Sayar (does Not have
to
Conservation of Energy = If NO collision
Z
Y
-
energy
origin)
T
be about
-
Do NOT use for
impact/collision Qs
T + V = y + Vav =
Vg + Ve
*
Set Datum and if :
y is o re
a = + Wy
y is below E-Wy
u = 0i + CoS30] + Sin30
corner permanently
hooks onto point
[kg m
a
Angular momentum (wrt 0 or 6) : .
1)(Hol , and Hot
-
Hx =
FxxWx -
[xyGy -
[xzWz 2) (HoA) ,
= (Ho) , Lot
3) Zon = ?
Hy FyxWx +
IyyWy [yzWz (Not principle axis)
-
Rotation
=
at
- -
:
4) [Ho1) , =
(Hot) = (Ho)2 Not ·
is first time z
-
Hz =
FzxWx -
1zywy + [zzwz * always find wand to save
-
Hy =
[xxwx -
xywy =
2xzw Z [Mx =
[xxx -
(Iy [z)wywz
- -
Exy(xy
-
wzWx)
-
[yz(wy2 (z) - =
2zx(xz +
0xwe)
Angular Moment Curt to point A) :
EMy =
zyxy
-
(2z -
[x)GzWx -
zyz(xz -
wxwy) -
[zx(wz -
wx) -
Exy(xz + Wywz)
Ma =
roXMVj + 40
[Mz =
zzXz -
(2x -
[y)wxwy -
2zx(Xx -
WyWz) -
2xy(wx wj) -
-
2yz(xy + wzwx)
·
