Download Physics 218 formula sheet and more Cheat Sheet Physics in PDF only on Docsity! Trigonometry and Vectors: sin 30◦ = cos 60◦ = 1 2 sin 36.9◦ ≈ cos 53.1◦ ≈ 3 5 sin 45◦ = cos 45◦ = 1√ 2 sin 53.1◦ ≈ cos 36.9◦ ≈ 4 5 sin 60◦ = cos 30◦ = √ 3 2 hadj = h cos θ = h sinφ hopp = h sin θ = h cosφ h2 = h2 adj + h2 opp tan θ = hopp hadj h hopp φ θ hadj Law of cosines: C2 = A2 +B2 − 2AB cos γ Law of sines: sinα A = sinβ B = sin γ C Bγ α C A β ~A = Axî+Ay ĵ +Az k̂ Â = ~A | ~A| ~A · ~B = AxBx +AyBy +AzBz = AB cos θ = A‖B = AB‖ ~A× ~B = (AyBz−AzBy )̂i+ (AzBx−AxBz)ĵ + (AxBy−AyBx)k̂ | ~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 (at n) = natn−1 d dt sin at = a cos at d dt cos at = −a sin at Integrals: ∫ t2 t1 f(t)dt = a n+1 (tn+1 2 − tn+1 1 ) if f(t) = atn, then { ∫ f(t)dt = a n+1 tn+1 + C (n 6= −1) ∫ sin at dt = −1 a cos at ∫ cos at dt = 1 a sin at Kinematics: translational rotational 〈~v〉 = ~r2−~r1 t2−t1 ~v = d~r dt 〈~a〉 = ~v2−~v1 t2−t1 ~a= d~v dt = d2~r dt2 ~r(t) = ~r0 + ∫ t 0 ~v(t′) dt′ ~v(t) = ~v0 + ∫ t 0 ~a(t′) dt′ 〈ω〉 = θ2−θ1 t2−t1 ω = dθ dt 〈α〉 = ω2−ω1 t2−t1 α= dω dt = d2θ dt2 θ(t) = θ0 + ∫ t 0 ω(t′) dt′ ω(t) = ω0 + ∫ t 0 α(t′) dt′ —– constant (linear/angular) acceleration only —– ~r(t) = ~r0 + ~v0t+ 1 2 ~at2 ~v(t) = ~v0 + ~at v2x = v2x,0 + 2ax(x− x0) (and similarly for y and z) ~r(t) = ~r0 + 1 2 (~vi + ~vf )t θ(t) = θ0 + ω0t+ 1 2 αt2 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 2 Mv2 W = ∫ ~F · d~r const−−−→ force ~F ·∆~r P = dW dt = ~F · ~v ~pcm = m1~v1 +m2~v2 + . . . = M~vcm ~J = ∫ ~Fdt = ∆~p ∑ ~Fext = M~acm = d~pcm dt ∑ ~Fint = 0 if ∑ Fext,x = 0, pcm,x = const ~τ = ~r × ~F and |~τ | = F⊥r = Fl Krot = 1 2 Itotω 2 W = ∫ τdθ const−−−−→ torque τ ∆θ P = dW dt = ~τ · ~ω ~L = ∑ ~r × ~p = I1~ω1 + I2~ω2 + . . . = Itot~ω ∑ ~τext = Itot~α = d~L dt ∑ ~τint = 0 if ∑ τext,z = 0, Lz = const —– Work-energy and potential energy —– W = ∆K Etot,i +Wother = Etot,f U = − ∫ ~F · d~r ; Ugrav = Mgycm ; Uelas = 1 2 k∆x2 Fx(x) = −dU(x)/dx ~F = −~∇U = − [ ∂U ∂x î+ ∂U ∂y ĵ + ∂U ∂z k̂ ] Constants/Conversions: g = 9.80 m/s 2 = 32.15 ft/s 2 (Earth, sea level) ≈ 10 m/s 2 ≈ 33 ft/s 2 G = 6.674× 10−11 N ·m2/kg2 1 mi = 1609 m 1 lb = 4.448 N 1 ft = 12 in ⇔ 0.454 kg (Earth, sea level) 1 in = 2.54 cm 1 rev = 360◦ = 2π radians Circular motion: arad = v2 R atan = d|~v| dt = Rα T = 2πR v s = Rθ vtan = Rω Relative velocity: ~vA/C = ~vA/B + ~vB/C ~vA/B = −~vB/A Forces: Newton’s: ∑ ~F = m~a, ~FB on A = −~FA on B Hooke’s: 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) Phys 218 — Challenge Exam Formulae
