Introductory Mechanics Formulas
g=GME
R2
E
= 9.81 m
s2= 9.81 N
kg ;RE= 6.4×106m ; ME= 6.0×1024kg ; G= 6.67 ×10−11 N·m2/kg2
Vector Principles
~
A=Axˆ
i+Ayˆ
j+Azˆ
k⇒(Ax, Ay, Az) ; ∆ ~
A≡~
Afinal −~
Ainitial ;~a ·~
b=ab cos θ;|~a ×~
b|=ab sin θ
Kinematics
~r =xˆ
i+yˆ
j;~v ≡d~r
dt ;~a ≡d~v
dt =d2~r
dt2;x(t) = x0+Zt
0
v(t0)dt0;v(t) = v0+Zt
0
a(t0)dt0;~vav ≡∆~r
∆t;~aav ≡∆~v
∆t
Constant Acceleration Kinematics
~v =~v0+~at ;~r =~r0+~v0t+1
2~at2;~r =~r0+1
2(~v0+~v)t;x=x0+1
2a(v2−v2
0)
Rotational Kinematics
θ≡s
r;ω≡dθ
dt ;α≡dω
dt =d2θ
dt2;vt=ds
dt =ωr ;at=d2s
dt2=αr ;ωav ≡∆θ
∆t;αav ≡∆ω
∆t
Uniform Circular Motion
acent =ar=v2
r=ω2r;T=2πr
v
Simple Harmonic Motion
x(t) = Acos(ωt +δ) ; f=1
T;ω= 2πf ;Tmass−spring = 2πrm
k;Tpend = 2πsL
g;Tphys−pend = 2πsI
mgD
Dynamics, Friction & Gravity
~
Fnet
m=~a ;~
FAB =−~
FBA ;|fs| ≤ µsN;|fk|=µkN;Fspring =−kx ;~
Fgrav
ab =−Gmamb
r2
ab
ˆrab ;Fgrav
earth,m=w=gm
Work, Energy & Momentum
Wby~
F=Z~
F·d~s =ZFxdx +ZFydy +ZFzdz ;K=1
2mv2; ∆U=−WBCF ;Fint,cons =−dU
dx
Ug=−GMm
r;Ug=mgy ;Usp =1
2kx2;Wext = ∆Esys = ∆K+ ∆Ug+ ∆Usp + ∆Echem + ∆Etherm ;f∆s= ∆Etherm
P≡dW
dt =~
F·~v ;v2f−v1f=−(v2i−v1i) ; ~p =m~v ;~
I=Z~
F dt = ∆~p ;XFext =d~
P
dt
Systems of Particles
~rcm =1
Mtot Xmi~ri;~rcm =R~rdm
Rdm
Rotational Dynamics
I=Xmir2
i;I=Zr2dm ;Ip=Icm +Mh2;K=1
2Iω2;Wrot =Zτdθ = ∆Krot ;P=dW
dt =τω ;~
L=~r ×~p
~τ =~r ×~
F;τ=r⊥F;X~τ =I ~α ;X~τ =d~
L
dt ;vcm =rω ;acm =rα ;~
L=I~ω
Moments of Inertia
cylindrical shell: Icm =MR2; disk: Icm = (1/2)MR2; rod: Icm = (1/12)ML2;
solid sphere: Icm = (2/5)MR2; hollow sphere: Icm = (2/3)MR2
Waves
y(x, t) = Asin(kx −ωt) ; ω= 2πf =2π
T;k=2π
λ;∂2y
∂x2=1
v2
∂2y
∂t2;vwave =fλ =ω
k;vwave on string =sF
µ
vair =sγP
ρ=rγRT
M;vsolid =sB
ρ;Pav =1
2µω2A2v;Iav =Pav
4πr2;
β= (10dB) log10 I
I0;I0= 10−12 W
m2;ysw =Asin(kx) cos(ωt) ; ∆Lconst =mλ ; ∆Ldest =m+1
2λ
Asin θ1+Asin θ2= 2Acos θ1−θ2
2sin θ1+θ2
2;Asin θ1−Asin θ2= 2Acos θ1+θ2
2sin θ1−θ2
2