Download Mechanics formula sheet and more Cheat Sheet Mechanical Engineering in PDF only on Docsity! 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 = Ax î + Ay ĵ + Az k̂ ⇒ (Ax, Ay, Az) ; ∆ ~A ≡ ~Afinal − ~Ainitial ; ~a ·~b = ab cos θ ; |~a × ~b| = ab sin θ Kinematics ~r = xî + yĵ ; ~v ≡ d~r dt ; ~a ≡ d~v dt = d2~r dt2 ; x(t) = x0 + ∫ t 0 v(t′)dt′ ; v(t) = v0 + ∫ t 0 a(t′)dt′ ; ~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) = A cos(ωt + δ) ; f = 1 T ; ω = 2πf ; Tmass−spring = 2π √ m k ; Tpend = 2π √ L g ; Tphys−pend = 2π √ I mgD Dynamics, Friction & Gravity ~Fnet m = ~a ; ~FAB = − ~FBA ; |fs| ≤ µsN ; |fk| = µkN ; F spring = −kx ; ~F grav ab = − Gmamb r2 ab r̂ab ; F grav earth,m = w = gm Work, Energy & Momentum W by~F = ∫ ~F · d~s = ∫ Fxdx + ∫ Fydy + ∫ Fzdz ; K = 1 2 mv2 ; ∆U = −WBCF ; Fint,cons = − dU dx Ug = − GMm r ; Ug = mgy ; Usp = 1 2 kx2 ; Wext = ∆Esys = ∆K + ∆Ug + ∆Usp + ∆Echem + ∆Etherm ; f∆s = ∆Etherm P ≡ dW dt = ~F · ~v ; v2f − v1f = −(v2i − v1i) ; ~p = m~v ; ~I = ∫ ~Fdt = ∆~p ; ∑ Fext = d~P dt Systems of Particles ~rcm = 1 Mtot ∑ mi~ri ; ~rcm = ∫ ~rdm ∫ dm Rotational Dynamics I = ∑ mir 2 i ; I = ∫ r2dm ; Ip = Icm + Mh2 ; K = 1 2 Iω2 ; Wrot = ∫ τdθ = ∆Krot ; P = dW dt = τω ; ~L = ~r × ~p ~τ = ~r × ~F ; τ = r⊥F ; ∑ ~τ = I~α ; ∑ ~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) = A sin(kx − ωt) ; ω = 2πf = 2π T ; k = 2π λ ; ∂2y ∂x2 = 1 v2 ∂2y ∂t2 ; vwave = fλ = ω k ; vwave on string = √ F µ vair = √ γP ρ = √ γRT M ; vsolid = √ B ρ ; Pav = 1 2 µω2A2v ; Iav = Pav 4πr2 ; β = (10dB) log10 ( I I0 ) ; I0 = 10−12 W m2 ; ysw = A sin(kx) cos(ωt) ; ∆Lconst = mλ ; ∆Ldest = ( m + 1 2 ) λ A sin θ1 + A sin θ2 = 2A cos ( θ1 − θ2 2 ) sin ( θ1 + θ2 2 ) ; A sin θ1 − A sin θ2 = 2A cos ( θ1 + θ2 2 ) sin ( θ1 − θ2 2 )
