Formula Sheet: PHYS 2310/2310H
Vectors and math:

A  Ax iˆ  Ay ˆj  Az kˆ

Aˆ  A / A

dA / dt  (dA / dt ) Aˆ  A(dAˆ / dt )
z
 
A  B  AB cos   Ax Bx  Ay By  Az Bz

A  A  Ax2  Ay2  Az2
k̂
 
A  B  ( Ay Bz  Az By )iˆ  ( Az Bx  Ax Bz ) ˆj  ( Ax By  Ay Bx )kˆ
 
A  B  AB sin 
d n
x  nx n 1
dx
iˆ
x
b  b 2  4ac
ax  bx  c  0 
x
2a
d
d
sin(ax  b)  a cos(ax  b)
cos(ax  b)  a sin(ax  b)
dx
dx
Geometry: Circle:
Sphere:
ĵ
y
2
Perimeter  2 R
Area   R
Surface area  4 R
4
Volume   R 3
3
Volume   R 2 h
2
Cylinder: Surface Area  2 Rh  2 R 2
2
1 revolution = 2 radians = 360
Conversion factors:
1 yard = 3 feet = 36 inches
1 mile = 1.609 km
1 inch = 2.54 cm
1 lb = 4.448 N
1 gallon = 3.788 liters
1 m3  1000 liters 1 L  atm 101.3 J
1 atm = 1.013  105 Pa  760 mm Hg
1 lb  454 g
Physical constants:
g  9.81 m/s 2
RE  6.38 106 m
R  8.314
1 food Cal = 1000 cal
G  6.67  1011 N  m 2 /kg 2
J
L  atm
 0.08206
mol  K
mol  K
vsound(air)  343 m/s
N A  6.022 1023 / mol
1 cal = 4.186 J


r
General kinematics: v average 
t
10−15
femto- (f)
10−12
pico- (p)
10−9
nano- (n)
10−6
micro- (μ)
10−3
milli- (m)
10−2
centi- (c)
103
kilo- (k)
106
mega- (M)
109
giga- (G)
1012
tera- (T)
M E  5.97 1024 kg
R
k
 1.3811023 J/K
NA
STP: 1 atm, 273.15 K

 dr
v
dt

v

aaverage 
t

 dv
a
dt
v x (t)  v0 x   ax (t ')dt '
x(t)  x0   vx (t ') dt '
t
t
0
0
Constant acceleration:
1
  
r  r0  v 0t  at 2
2
1
x  x0  v 0 x t  ax t 2
2
  
v  v 0  at
 
v 2  v 02  2a  r
v x  v 0 x  axt
v x  v 02x  2ax x
2
1
x  x0  (v0 x  vx )t
2
Circular motion:
arad 
v2
R
atan 
dv
dt
 

a  arad  atan
Relative motion:



rA rel. to C  rA rel. to B  rB rel. to C



vA rel. to C  vA rel. to B  vB rel. to C
Forces:


 F  ma



Fgrav ( w)  mg


FAB   FBA
Constant v: v 
2
2
a  arad
 atan
2 R
T



aA rel. to C  aA rel. to B  aB rel. to C
fs  s N
FHooke  k x
fk  k N
Work and energy:
 
W12   12 F  dr
dW  
 F v
dt

r 



U (r )  U (r0 )    F  dr
Pave 
W
t
1
p2
K  mv 2 
2m
2
W  F x cos 
Wconservative  U
P 
Wnet  K
Ugrav  mgy  C
U elas 
1 2
kx  C
2
r0
E  K U
E  K  U  Wnon-conservative (E  0 when only conservative forces do work)
Momentum, impulse. Systems of particles:





p  mv
J  p   F dt


rCM
m r

m
i i
i


v CM
i
 mv

m
i i
i
i
i
i



ptotal   pi  mtotal vCM
i
t  t2  t1


aCM
m a

m
i i
i



p J
Fave 

t t
K lab  K CM  K relative to CM
i
i


dptotal

Fnet 
 mtotal aCM
dt
Collisions between two masses in one dimension:

When F  0, p
net
total,i

 ptotal,f

mA vA1x  mBvB1x  mA vA2 x  mBvB2 x
Elastic collisions between two masses in one dimension:
vA2x 
(mA  mB )vA1x  2mBvB1x
mA  mB
(mA  mB )
vA1x
mA  mB
vB2x 
vA1x  vB1x  (vA 2 x  vB2 x )
vA2x 
vB2x 
(mB  mA )vB1x  2mA vA1x
mA  mB
2mA
vA1x
mA  mB
Rigid-body motion: kinematics and dynamics:
z 
d
dt

d
dt
z 
dz
dt

dz
d

dt
dt
1 2

f

Constant : T 
1
Constant α:    0  0 z t   z t 2
2
 z  0 z   z t
K total  K translation  K rotation
K translation 
I   mi ri 2
  
  r F
I  I CM  md 2
i

dL
 net 
dt
 z2  02z  2 z (   0 )
1 2
mvCM
2
2
For rotation about a fixed axis:
Pave 
1
P 
2
MR 2
5
2
I hollow sphere  MR 2
3
I solid sphere 
L
2 r 3 2
T
GM
g  9.81 m/s 2
Lz  I  z
 
Pave    ave
dW
dt
1
MR 2
2
I hollow cylinder  MR 2
with thin walls
b
1
ML2
12
 
P   
I rectangle 
1
M  a 2  b2 
12
a

F  0
Gravitation:

Mm
FNewton  G 2
r
  
Lrp
I solid cylinder 
I rod 

1 2
I
2


 net  I
K rotation 
W   z 
W    z d
W
t
1
2
   0  (0 z  z )t


(When  net  0, Ltotal,i  Ltotal,f )

Statics:
atan  R
v  R
s  R
  0
g G
M
R2
U  G
Mm
r
v circular orbit 
GM
r
 
r  v  constant
G  6.67 1011 N  m 2 /kg 2
RE  6.38 106 m
M E  5.97 1024 kg
vescape 
2GM
r
Simple harmonic motion:
d 2x
2x  0
2
dt
x  A cos( t   )
FHooke   k x

k
m

Static Fluids:
p
F
A
p   g h
v   A sin(t   )


I

a   A 2 cos(t   )
mgd
I
T
1 2


f
g
l

dm
dV
1 atm  1.013  105 Pa  760 mm Hg
E
1
1
mv 2  kx 2   kA2

2
2
Temperature and heat:
TK  TC  273.15
9
TF  TC  32
5
Q  mcT
Ideal gas:
pV  nRT  NkT
k
n
Nmmolecule
N

NA
M
N A  6.022 1023 / mol R  8.314
J
L  atm
 0.08206
mol  K
mol  K
R
 1.381 1023 J/K
NA
Thermodynamics:
p
F
A
U  Q  W
Thermodynamic processes:
Isochoric: constant V
Isobaric: constant p
W 
Vf
Vi
pdV
Isothermal: constant T
W  pV
Adiabatic: Q  0