Trigonometry and Vectors:
sin 30◦= cos 60◦=1
2sin 36.9◦≈cos 53.1◦≈3
5
sin 45◦= cos 45◦=1
√2sin 53.1◦≈cos 36.9◦≈4
5
sin 60◦= cos 30◦=√3
2
hadj =hcos θ=hsin φ
hopp =hsin θ=hcos φ
h2=h2
adj +h2
opp
tan θ=hopp
hadj
hhopp
φ
θ
hadj
Law of cosines: C2=A2+B2−2AB cos γ
Law of sines: sin α
A=sin β
B=sin γ
C
B
γ
α
C
A
β
~
A=Axˆ
i+Ayˆ
j+Azˆ
kˆ
A=~
A
|~
A|
~
A·~
B=AxBx+AyBy+AzBz=AB cos θ=AkB=ABk
~
A×~
B= (AyBz−AzBy)ˆ
i+ (AzBx−AxBz)ˆ
j+ (AxBy−AyBx)ˆ
k
|~
A×~
B|=AB sin θ=A⊥B=AB⊥(direction via right-hand rule)
Quadratic:
ax2+bx +c= 0 ⇒x1,2=−b±√b2−4ac
2a
Derivatives: d
dt (atn) = natn−1
d
dt sin at =acos at
d
dt cos at =−asin at
Integrals: Rt2
t1f(t)dt =a
n+1 (tn+1
2−tn+1
1)
if f(t) = atn, then (Rf(t)dt =a
n+1 tn+1 +C
(n6=−1)
Rsin at dt =−1
acos at
Rcos at dt =1
asin at
Kinematics:
translational rotational
h~vi=~r2−~r1
t2−t1~v =d~r
dt
h~ai=~v2−~v1
t2−t1~a =d~v
dt =d2~r
dt2
~r(t) = ~r0+Rt
0~v(t′)dt′
~v(t) = ~v0+Rt
0~a(t′)dt′
hωi=θ2−θ1
t2−t1ω=dθ
dt
hαi=ω2−ω1
t2−t1α=dω
dt =d2θ
dt2
θ(t) = θ0+Rt
0ω(t′)dt′
ω(t) = ω0+Rt
0α(t′)dt′
—– constant (linear/angular) acceleration only —–
~r(t) = ~r0+~v0t+1
2~at2
~v(t) = ~v0+~at
v2
x=v2
x,0+ 2ax(x−x0)
(and similarly for yand z)
~r(t) = ~r0+1
2(~vi+~vf)t
θ(t) = θ0+ω0t+1
2αt2
ω(t) = ω0+αt
ω2
f=ω2
0+ 2α(θ−θ0)
θ(t) = θ0+1
2(ωi+ωf)t
Energy and Momenta:
translational rotational
K=1
2Mv2
W=R~
F·d~r const
−−−→
force
~
F·∆~r
P=dW
dt =~
F·~v
~pcm =m1~v1+m2~v2+...
=M~vcm
~
J=R~
F dt = ∆~p
P~
Fext =M~acm =d~pcm
dt
P~
Fint = 0
if PFext,x = 0, pcm,x =const
~τ =~r ×~
Fand |~τ|=r⊥F=F⊥r
Krot =1
2Itotω2
W=Rτdθ const
−−−→
torque τ∆θ
P=dW
dt =~τ ·~ω
~
L=P~r ×~p
=I1~ω1+I2~ω2+...
=Itot~ω
P~τext =Itot~α =d~
L
dt
P~τint = 0
if Pτext,z = 0, Lz=const
—– Work-energy and potential energy —–
W= ∆K W =−∆U Etot,i +Wother =Etot,f
U(r) = −R~
F·d~r ;Ugrav =Mgycm ;Uelas =1
2k∆x2
Fx(x) = −dU(x)/dx ~
F=−~
∇U=−∂U
∂x ˆ
i+∂U
∂y ˆ
j+∂U
∂z ˆ
k
Constants/Conversions:
g=9.80 m/s2= 32.15 ft/s2(Earth, sea level)
≈10 m/s2≈33 ft/s2
G=6.674 ×10−11 N·m2/kg21mi = 1609 m
1lb = 4.448 N1ft = 12 in
⇔0.454 kg (Earth, sea level) 1in = 2.54 cm
1rev = 360◦= 2πradians
Circular motion: arad =v2
Ratan =d|~v|
dt =Rα
T=2πR
vs=Rθ vtan =Rω
Relative velocity: ~vA/C =~vA/B +~vB/C
~vA/B =−~vB/A
Forces:
Newton’s Law: P~
F=m~a, ~
FBon A=−~
FAon B
Hooke’s Law: Fx=−k∆x
friction: |~
fs| ≤ µs|~n|,|~
fk|=µk|~n|
Centre-of-mass:
~rcm =m1~r1+m2~r2+... +mn~rn
m1+m2+...+mn
(and similarly for ~v and ~a)
Gravity: ~
Fgrav =−GM1M2
r2ˆr Ugrav =−GM1M2
r
Kepler’s Laws:
1st:
2nd:~r ×~v =constant
3rd:T=2πa3/2
√GM
Phys 206 — Comprehensive Exam Formulae
