Physics 121, Formula Sheet
Final Exam
Geometry/Trigonometry:
1
3 sin ( 30° ) =
2
1
cos ( 45° ) =
2 sin ( 45° ) =
2
1
cos ( 60° ) =
sin ( 60° ) =
2
cos ( 30° ) =
1
2
1
2
2
1
3
2
tan ( 30° ) =
1
3
3
tan ( 45° ) = 1
tan ( 60° ) = 3
$1
'
cos & ! " # ) = sin (# )
%2
(
$1
'
sin & ! " # ) = cos (# )
%2
(
cos ( 2# ) = 1 " 2 sin 2 (# )
sin ( 2# ) = 2 sin (# ) cos (# )
circle
circumference
2! r
(surface) area
!r2
volume
sphere
4! r 2
4 3
!r
3
Integrating and Differentiating:
d(x n )
= nx n !1
dx
x n +1
n
x
dx
=
"
n +1
Linear Motion in One Dimension (general):
dx
dt
dv d 2 x
a=
=
dt dt 2
v=
- 1 -
Physics 121, Formula Sheet
Final Exam
Linear Motion in One Dimension (special case):
a(t) = a = constant
v(t) = v0 + at
x(t) = x0 + v0t +
1 2
at
2
Linear Motion in Two/Three Dimensions:
!
! dr
v=
dt
!
!
! dv d 2 r
a=
=
dt dt 2
Circular Motion:
v2
aR =
r
dv
atan =
dt
Force Laws:
!
!
! F = ma
Newton's Second Law of Motion
!
!
F12 = " F21
Newton's Third Law of Motion
i
i
Friction:
Fs ! µ s N
Fk = µ k N
FD = "bv
Static Friction
Kinetic Friction
Dragg Force
- 2 -
Physics 121, Formula Sheet
Final Exam
Newton’s Gravitational Law:
!
mm
F12 = !G 12 2 r̂21
r 21
Kepler’s Third Law (Law of Periods):
2
! T1 $
! r1 $
#" T &% = #" r &%
2
2
3
Work Done by a Force:
! !
W = F•d
!
b !
W = ! F • dl
a
Constant Force
Variable Force
Translation Kinetic Energy:
K=
1 2
mv
2
Work-Energy Theorem:
W = !K
Potential Energy:
!
2 !
!U = U 2 " U1 = " # F • dl = "W
1
General Definition of Potential Energy
!
dU
dU
dU
F="
x̂ "
ŷ "
ẑ
dx
dy
dz
U(h) = mgh
Gravitational Potential Energy (Close to the Surface)
U(r) = !G
U(x) =
mM E
r
1 2
kx
2
Gravitational Potential Energy (r>rE )
Spring with Spring Constant k
- 3 -
Physics 121, Formula Sheet
Final Exam
Conservation of Energy:
!U + !K = 0
!U + !K = WNC
Conservation of Mechanical Energy
Conservation of Energy
Power:
P=
dW
dt
Linear Momentum and Newton’s Second Law:
!
!
p = mv
!
!
dp
= !F
dt
Collision Impulse:
!
J=
!
t2
t1
!
!
!
!
Fdt = p f " pi = #p
Elastic Collisions in One Dimension:
" m ! m2 %
" 2m2 %
v1 ' = v1 $ 1
+ v2 $
'
# m1 + m2 &
# m1 + m2 '&
" 2m1 %
" m ! m2 %
v2 ' = v1 $
+ v2 $ 1
'
# m1 + m2 &
# m1 + m2 '&
Center of Mass:
!
!
rcm =
!m r
i i
i
!m
i
i
1
!
rcm =
M
"
!
rdm
- 4 -
Physics 121, Formula Sheet
Final Exam
Motion of the Center of Mass:
!
!
Macm = ! Fi
Rocket Equations:
!
!
dv
! dM
M
= ! Fext + vrel
dt
dt
Marocket = RU 0
"M %
v f = vi + u ln $ i '
# Mf &
First Rocket Equation
Second Rocket Equation
Angular Variables:
Definition Linear Variable
Angular Position
!
Angular Velocity
"=
Angular Acceleration # tan
d!
dt
d 2!
= 2
dt
l = R!
v = R"
atan = R# tan
Equations of Motion for Constant Acceleration:
Rotationional Motion
Linear Motion
Acceleration
! (t) = !
a(t) = a
Velocity
" (t) = " 0 + ! t
v(t) = v0 + at
Position
1
# = # 0 + " 0t + ! t 2
2
x(t) = x0 + v0t +
- 5 -
1 2
at
2
Physics 121, Formula Sheet
Final Exam
Moment of Inertia:
I = ! mr ri 2 (individual point masses)
Moment of Inertia:
i
I=
"
r 2 dm (continuous mass distribution)
Voume
Parallel-axis Theorem
I = I cm + Mh 2
Perpendicular-axis Theorem
Iz = Ix + Iy
Torque:
Definition:
! =r "F
Newton's Second Law for Rotational Motion: ! = I#
Angular Momentum:
Definition:
L=r!p
Rotating rigid object:
L = I"
Relation between torque and angular momentum:
dL
= $#
dt
Rotational Energy:
1 2
I!
2
Kinetic energy:
K=
Work:
W = $ " d#
Power:
P = "!
Work-Energy Theorem: W = %K =
1
1
I! f 2 & I! i 2
2
2
Precession:
!=
Mgrcm
L
- 6 -
Physics 121, Formula Sheet
Final Exam
Moments of inertia of various objects of uniform composition.
- 7 -
Physics 121, Formula Sheet
Final Exam
Conditions for Equilibrium:
!F
=0
!"
x
=0
!F
=0
!"
y
=0
!F
=0
!"
z
=0
x
y
z
Hooke’s Law:
F = k!L
Stress:
stress = force/area = F/A
Strain:
strain = change in length / original length = ΔL/L0
Young’s Modulus E:
E = stress/strain
Simple Harmonic Motion:
x(t) = A cos(! t + " )
A = Amplitude
! = angular frequency
" = phase
Force Requirement: F(x) = #m! 2 x
Period:
T = 2$ / !
Frequency:
f = 1 / T = ! / 2$
Definition:
The Physical Pendulum:
T = 2!
I
mgh
- 8 -
Physics 121, Formula Sheet
Final Exam
Damped Harmonic Motion (damping force = -bv):
Solution: x(t) = Ae!" t cos(# t)
A = Amplitude
k
b2
#=
!
m 4m 2
b
"=
2m
Forced Harmonic Motion (external force Fext = F0 cos(ω t) and damping force = -bv):
Solution: x(t) = A0 cos(! t + "0 )
F0
A0 =
m
(!
2
# ! 02
)
2
+
b 2! 2
m2
$
'
2
2)
&
! #!
"0 = tan #1 & 0
)
& ! $& b ') )
% % m( (
!0 =
k
m
Thermal Expansion:
!L = " L0 !T
Linear Expansion
!V = #V0 !T
# $ 3"
Volume Expansion
Ideal Gas Law:
PV = nRT
PV = NkT
Average Translational Kinetic Energy for an Ideal Gas:
K=
3
kT
2
- 9 -
Physics 121, Formula Sheet
Final Exam
The Maxwell Distribution:
3
2
1 mv
" m % 2 2 ( 2 kT
f ( v ) = 4! N $
v
e
# 2! kT '&
Mean free path in a gas:
lM =
1
4! r ( N / V )
2
Specific Heat c:
Q = mc!T
Molar Specific Heats for Gases:
Q = nCV !T
Constant Volume
Q = nCP !T
Constant Pressure
CP " CV = R
CV =
3
R
2
Ideal Monatomic Gas
Latent Heat L:
Q = mL
First Law of Thermodynamics:
!U = Q " W
Adiabatic Expansion of a Gas:
PV ! = constant
- 10 -
Physics 121, Formula Sheet
Final Exam
Work Done during Volume Changes of an Ideal Gas:
W = nRT ln
VB
VA
" V %
W = nRTB $ 1 ! A '
# VB &
Isothermal Process
Isobaric Process
Heat Transfer:
!Q
T " T2
= kA 1
!t
l
Efficiency of a Heat Engine:
e=
W
QH
Coefficient of Performance of Refrigerators and Air Conditioners:
QL
W
CP =
Coefficient of Performance of Heat Pumps:
QH
W
CP =
Carnot Efficiency:
e = 1!
TL
TH
Entropy:
dS =
dQ
T
- 11 -