EN PH 131 - Final Exam Formula Sheet

EN PH 131 - Final Exam Formula Sheet Trigonometry opp sin θ = hyp cos θ = adj hyp tan θ = sin(A ± B) = sin A cos B ± cos A sin B opp sin θ = adj cos θ a2 + b2 = c2 cos(A ± B) = cos A cos B ∓ sin A sin B Quadratic Equations a x 2 + bx + c = 0 for a ≠ 0 has solutions: x = −b ± b 2 − 4a c 2a Vectors A ≡ A⃗ i ̂ = (1,0,0) , j ̂ = (0,1,0) , k̂ = (0,0,1) (3D) , i ̂ = (1,0) , j ̂ = (0,1) (2D) Note: A ⃗ ⋅ A ⃗ = Ax2 + Ay2 + Az2 = ∣ A ⃗ ∣2 A ⃗ ⋅ B ⃗ ≡ Ax Bx + Ay By + Az Bz = A B cos θAB A ⃗ × B ⃗ = (Ay Bz − Az By ) i ̂ + (Az Bx − Ax Bz ) j ̂ + (Ax By − Ay Bx )k̂ A ⃗ × B ⃗ = A B sin θ B ⃗× A ⃗ = − A ⃗× B ⃗ AB and i ̂ × j ̂ = k̂ plus cyclic permutations. Kinematics (General) Δr ⃗ Δr ⃗ d r ⃗ ⃗ ≡ v ⃗ ≡ lim = 2/3D: v av Δt dt Δt→0 Δt 1D: vav ≡ Δx Δt v ≡ lim Δx dx = Δt dt ⃗ ≡ 2/3D: a av 1D: aav ≡ Δv Δt a ≡ lim Δv dv = Δt dt Δv⃗ Δt a ⃗ ≡ lim Δt→0 Δv⃗ d v⃗ = Δt dt dx dv Time independent: v = , a= dt dt Δt→0 Δt→0 d dx d2x a= = dt ( dt ) dt 2 ⟹ a d x = vd v t t x v v 2 v02 a(x)d x = vd v = − Integral relationships 1D: v(t) = v0 + a(t)dt , x (t) = x0 + v(t)dt , ∫0 ∫0 ∫x ∫v 2 2 0 0 Separation of variables (general form): If d y /d x = f (x)g(y) then Normal-Tangential: y ∫y d y /g(y) = d ∣ v⃗∣ v2 at = , a rad = an = acp = , a =∣ a⃗∣= dt r 1 + ( dx ) [ ] dy Radius of curvature: y = y(x) ⟹ r (x) = 0 x ∫x f (x)d x 0 at2 + an2 , at d s = vd v 3 2 2 d 2 y /d x 2 , circular paths: s = r θ Kinematics 1D (const a = ac) 1 2 x (t) = x0 + v0x t + ax t 2 vx (t) = v0x + ax t vx2 = v0x + 2ax (x − x0 ) 2 v0 + v x (t) = x0 + vav,x t = x0 + t ( 2 ) ⃗ = v A/B ⃗ + v B/C ⃗ ⃗ = − v A/B ⃗ Relative motion (Galilean/non-relativistic): v A/C v B/A ⃗ = F net Newton’s Second Law: Fnet,x = Cartesian: ∑ Normal/Radial-Tangential: Fnet,t = ∑ Fext,t ∑ d ∣ v⃗∣ = m at = m dt ⃗ = ma ⃗ F ext Fext,x = m ax Fnet,y = ∑ Fext,y = m a y v2 Fnet,rad = F = m a rad = m ∑ ext,rad r Force Prescriptions G m1m 2 r2 ⃗ ⃗ Normal forces ( N ), and tensional forces ( T ) are constraint forces that have no predetermined formula, but must always be solved for. Frictional forces: fk = μk N , ( f k⃗ = − μk ∣ N ⃗ ∣ ut̂ ) , fs ≤ μs N ⃗ = − k x ⃗ (3D) fres(x) = − k x (1D) , f res Hooke’s ‘Law’: Gravitational/Weight: W = mg , Fgrav = Work/Energy constant force, straight-line displacement: 1D var force/displacement: W= xf ∫x W ≡ F ⃗ ⋅ s ⃗ = ∣ F ⃗ ∣ ∣ s ⃗ ∣ cos ϕ F(x)d x 2D/3D var force: W= 0 P2 ∫P F ⃗⋅ d r⃗ = 1 P2 ∫P F cos ϕd s 1 W dW 1 Kinetic Energy: K ≡ mv 2 Power: Pav = , P= = F ⃗⋅ v ⃗ 2 Δt dt 1 1 Wtotal = WF = ΔK = mvf2 − mv02 Work-Energy Theorem: net 2 2 Work done by gravity: Wgrav = mg y0 − mg yf = − ΔUgrav 1 1 Work done by linear spring: Wspring = k x02 − k xf2 = − ΔUspring 2 2 Wc,F = − ΔUF Conservative forces: (1D) Fx = − dUF (x)/d x 1 Ugrav(y) = mg y , Uspring(x) = k x 2 Potential Energies: 2 Mechanical energy: E =K+U Conservation of mechanical energy (Wnc = 0): E = const ⇔ ΔE = 0 ⇔ K0 + U0 = Kf + Uf Conservation of energy (general): Wnc = ΔE = ΔK + ΔU ( = − ΔUint ) Impulse/Momentum impulse: J F⃗ ≡ momentum: p ⃗ ≡ m v ⃗ ⃗ = Δp⃗ Impulse-momentum Theorem: J net tf ∫t ⃗ dt F (t) 0 i.e. p 1⃗ + t2 ∫t 1 ⃗ (t) dt = p ⃗ F net 2 F = 0 THEN p = p (where n = x, y, z, etc.). ∑ ext,sys,n ∑ i1,n ∑ i2,n i i vB2n − vA2n Coefficient of restitution: e = where n is the normal direction to the plane of tangency. vA1n − vB1n Conservation of momentum: IF Particle Systems: M r cm ⃗ = ∑ i ⃗ = mi r i⃗ , M v cm ∑ i mi v i⃗ , ∑ ⃗ =Ma⃗ = F ext cm ∑ i mi a i⃗ , Ugrav = mg ycm Rotational Kinematics/Dynamics d ωz dθ 2D Angular acceleration: αz = (i.e. about z axis) dt dt 1 ωz(t) = ω0 + αt θ (t) = θ0 + ω0t + αt 2 ω 2 = ω02 + 2α(θ − θ0 ) Const αz kinematics: αz = α 2 2π ω= Connection formulae: s = r θ , vt = r ω , at = r α , an = arad = ω 2r In UCM: T | τ ⃗ | = rF sin ϕr& Torque: τ ⃗ ≡ r ⃗ × F ⃗ = rF = Fl ⊥ ⃗ F⃗ 2D Angular velocity: ωz = I= Moment of Inertia: (discrete): ∑ 1 2 = Iω 2 Rotational work: Wrot = τz d θ ∫ Angular Momentum: L ⃗ = r ⃗ × p ⃗ IF ∑ i miri2 τz = Iαz Rotational analogue of Newton II: Rotational Kinetic Energy: Krot ∑ (continuous 1D): 1 1 2 mvcm + Icm ω 2 2 2 Roll without slip: vcm = vt , acm = at General kinetic energy: K = ; (Newton II): τnet ⃗ = τext,sys ⃗ = 0 ⃗ THEN L ⃗ = const, i.e. L 0⃗ = L f⃗ . d L⃗ dt ; Lpt = mvr sin ϕr& ⃗ p ⃗ , Lz = Iωz Moments of Inertia for Common Uniform Continuous Distributions: thin uniform rod, axis about center: thin uniform rod, axis about one end: uniform solid cylinder/disk, axis through center: uniform hollow cylinder or hoop axis through center: uniform solid sphere, axis through center: uniform hollow sphere, axis through center: Parallel axis theorem: I = Icm + m d 2 I = x 2d m ∫ ML 2 /12 ML 2 /3 MR 2 /2 MR 2 2MR 2 /5 2MR 2 /3