homework 09 – BAUTISTA, ALDO – Due: Oct 30 2006, 4:00 am
1
Mechanics  Basic Physical Concepts
Math:
Circle: 2
π r
,
π r
2
; Sphere: 4
π r
2
, (4
/
3)
π r
3
Quadratic Eq.:
a x
2
+
b x
+
c
= 0,
x
=

b
±
√
b
2

4
a c
2
a
Cartesian and polar coordinates:
x
=
r
cos
θ, y
=
r
sin
θ
,
r
2
=
x
2
+
y
2
,
tan
θ
=
y
x
Trigonometry:
cos
α
cos
β
+ sin
α
sin
β
= cos(
α

β
)
sin
α
+ sin
β
= 2 sin
α
+
β
2
cos
α

β
2
cos
α
+ cos
β
= 2 cos
α
+
β
2
cos
α

β
2
sin 2
θ
= 2 sin
θ
cos
θ,
cos 2
θ
= cos
2
θ

sin
2
θ
1

cos
θ
= 2 sin
2
θ
2
,
1 + cos
θ
= 2 cos
2
θ
2
Vector algebra:
~
A
= (
A
x
, A
y
) =
A
x
ˆ
ı
+
A
y
ˆ
Resultant:
~
R
=
~
A
+
~
B
= (
A
x
+
B
x
, A
y
+
B
y
)
Dot:
~
A
·
~
B
=
A B
cos
θ
=
A
x
B
x
+
A
y
B
y
+
A
z
B
z
Cross product:
ˆ
ı
×
ˆ
=
ˆ
k
,
ˆ
×
ˆ
k
= ˆ
ı
,
ˆ
k
×
ˆ
ı
= ˆ
~
C
=
~
A
×
~
B
=
fl
fl
fl
fl
fl
fl
ˆ
ı
ˆ
ˆ
k
A
x
A
y
A
z
B
x
B
y
B
z
fl
fl
fl
fl
fl
fl
C
=
A B
sin
θ
=
A
⊥
B
=
A B
⊥
,
use right hand rule
Calculus:
d
dx
x
n
=
n x
n

1
,
d
dx
ln
x
=
1
x
,
d
dθ
sin
θ
= cos
θ
,
d
dθ
cos
θ
=

sin
θ
,
d
dx
const = 0
Measurements
Dimensional analysis:
e.g.
,
F
=
m a
→
[
M
][
L
][
T
]

2
,
or
F
=
m
v
2
r
→
[
M
][
L
][
T
]

2
Summation:
∑
N
i
=1
(
a x
i
+
b
) =
a
∑
N
i
=1
x
i
+
b N
Motion
One dimensional motion:
v
=
d s
dt
,
a
=
d v
dt
Average values:
¯
v
=
s
f

s
i
t
f

t
i
,
¯
a
=
v
f

v
i
t
f

t
i
One dimensional motion (constant acceleration):
v
(
t
) :
v
=
v
0
+
a t
s
(
t
) :
s
= ¯
v t
=
v
0
t
+
1
2
a t
2
,
¯
v
=
v
0
+
v
2
v
(
s
) :
v
2
=
v
2
0
+ 2
a s
Nonuniform acceleration:
x
=
x
0
+
v
0
t
+
1
2
a t
2
+
1
6
j t
3
+
1
24
s t
4
+
1
120
k t
5
+
1
720
p t
6
+
. . .
, (jerk, snap,
. . .
)
Projectile motion:
t
rise
=
t
fall
=
t
trip
2
=
v
0
y
g
h
=
1
2
g t
2
fall
, R
=
v
ox
t
trip
Circular:
a
c
=
v
2
r
,
v
=
2
π r
T
,
f
=
1
T
(Hertz=s

1
)
Curvilinear motion:
a
=
q
a
2
t
+
a
2
r
Relative velocity:
~v
=
~v
0
+
~u
Law of Motion and applications
Force:
~
F
=
m~a, F
g
=
m g,
~
F
12
=

~
F
21
Circular motion:
a
c
=
v
2
r
, v
=
2
π r
T
= 2
π r f
Friction:
F
static
≤
μ
s
N
F
kinetic
=
μ
k
N
Equilibrium (concurrent forces):
∑
i
~
F
i
= 0
Energy
Work (for all F):
Δ
W
=
W
AB
=
W
B

W
A
F
k
s
=
Fs
cos
θ
=
~
F
·
~s
→
R
B
A
~
F
·
d~s
(in Joules)
Effects due to work done:
~
F
ext
=
m~a

~
F
c

~
f
nc
W
ext

A
→
B
=
K
B

K
A
+
U
B

U
A
+
W
diss

A
→
B
Kinetic energy:
K
B

K
A
=
R
B
A
m~a
·
d~s
,
K
=
1
2
m v
2
K (conservative
~
F
):
U
B

U
A
=

R
B
A
~
F
·
d~s
U
gravity
=
m g y
,
U
spring
=
1
2
k x
2
From
U
to
~
F
:
F
x
=

∂ U
∂x
,
F
y
=

∂ U
∂y
,
F
z
=

∂ U
∂z
F
gravity
=

∂ U
∂y
=

m g
,
F
spring
=

∂ U
∂x
=

k x
Equilibrium:
∂ U
∂x
= 0,
∂
2
U
∂x
2
>
0 stable,
<
0 unstable
Power:
P
=
d W
dt
=
F v
k
=
F v
cos
θ
=
~
F
·
~v
(Watts)
Collision
Impulse:
~
I
= Δ
~
p
=
~
p
f

~
p
i
→
R
t
f
t
i
~
F dt
Momentum:
~
p
=
m~v
Twobody:
x
cm
=
m
1
x
1
+
m
2
x
2
m
1
+
m
2
p
cm
≡
M v
cm
=
p
1
+
p
2
=
m
1
v
1
+
m
2
v
2
F
cm
≡
F
1
+
F
2
=
m
1
a
1
+
m
2
a
2
=
M a
cm
K
1
+
K
2
=
K
*
1
+
K
*
2
+
K
cm
Twobody collision:
~
p
i
=
~
p
f
= (
m
1
+
m
2
)
~v
cm
v
*
i
=
v
i

v
cm
,
v
0
i
=
v
*0
i
+
v
cm
Elastic:
v
1

v
2
=

(
v
0
1

v
0
2
),
v
*0
i
=

v
*
i
,
v
0
i
= 2
v
cm

v
i
Many body center of mass:
~
r
cm
=
∑
m
i
~r
i
∑
m
i
=
R
~r dm
R
m
i
Force on cm:
~
F
ext
=
d~p
dt
=
M~a
cm
,
~
p
=
∑
~
p
i
Rotation of RigidBody
Kinematics:
θ
=
s
r
,
ω
=
v
r
,
α
=
a
t
r
Moment of inertia:
I
=
∑
m
i
r
2
i
=
R
r
2
dm
I
disk
=
1
2
M R
2
,
I
ring
=
1
2
M
(
R
2
1
+
R
2
2
)
I
rod
=
1
12
M ‘
2
,
I
rectangle
=
1
12
M
(
a
2
+
b
2
)
I
sphere
=
2
5
M R
2
,
I
spherical shell
=
2
3
M R
2
I
=
M
(Radius of gyration)
2
,
I
=
I
cm
+
M D
2
Kinetic energies:
K
rot
=
1
2
I ω
2
,
K
=
K