International Physics Olympiad (IPhO) Formulas Cheat Sheet, Cheat Sheet of Complex Numbers Theory

Download International Physics Olympiad (IPhO) Formulas Cheat Sheet and more Cheat Sheet Complex Numbers Theory in PDF only on Docsity! Formulas for IPhO Version: July 4, 2018 I Mathematics 1. Taylor series (truncate for approximations): F (x) = F (x0) + ∑ F (n)(x0)(x− x0)n/n! Special case — linear approximation: F (x) ≈ F (x0) + F ′(x0)(x− x0) Some examples for |x|  1: sin x ≈ x, cosx ≈ 1− x2/2, ex ≈ 1 + x ln(1 + x) ≈ x, (1 + x)n ≈ 1 + nx 2. Perturbation method: find the solution itera- tively using the solution to the ”non-perturbed” (directly solvable) problem as the 0th approxi- mation; corrections for the next approximation are calculated on the basis on the previous one. 3. Solution of the linear differential equation with constant coefficients ay′′ + by′ + cy = 0: y = A exp(λ1x) +B exp(λ2x), where λ1,2 is the solution of the characteristic equation aλ2 + bλ + c = 0 if λ1 6= λ2. If the solution of the characteristic equation is com- plex, while a, b and c are real numbers, then λ1,2 = γ ± iω and y = Ceγx sin(ωx+ ϕ0). 4. Complex numbers z = a+ bi = |z|eiϕ, z̄ = a− ib = |z|e−iϕ |z|2 = zz̄ = a2 + b2, ϕ = arg z = arcsin b|z| Rez = (z + z̄)/2, Imz = (z − z̄)/2 |z1z2| = |z1||z2|, arg z1z2 = arg z1 + arg z2 eiϕ = cosϕ+ i sinϕ cosϕ = e iϕ+e−iϕ 2 , sinϕ = eiϕ−e−iϕ 2i 5. Cross and dot products of vectors are dis- tributive: a(b+ c) = ab+ ac. ~a ·~b = ~b · ~a = axbx + ayby + . . . = ab cosϕ |~a×~b| = ab sinϕ; ~a×~b = −~b× ~a ⊥ ~a,~b ~a×~b = (aybz − byaz)~ex + (azbx− bzax)~ey + . . . ~a× [~b× ~c] = ~b(~a · ~c)− ~c(~a ·~b). Mixed prod. (volume of parallelep. def. by 3 vec.): (~a,~b,~c) ≡ (~a · [~b× ~c]) = ([~a×~b] · ~c) = (~b,~c,~a). 6. Cosine and sine laws: c2 = a2 + b2 − 2ab cosϕ a/ sinα = b/ sin β = 2R 7. sin(α± β) = sinα cosβ ± cosα sin β cos(α± β) = cosα cosβ ∓ sinα sin β tan(α± β) = (tanα+ tan β)/(1∓ tanα tan β) cos2 α = 1+cos 2α2 , sin 2 α = 1−cos 2α2 cosα cosβ = cos(α+β)+cos(α−β)2 , . . . cosα+ cosβ = 2(cos α+β2 + cos α−β 2 ), . . . . 8. An angle inscribed in a circle is half of the central angle that subtends the same arc on the circle. Conclusions: hypotenuse of a right triangle is the diameter of its circumcircle; if the angles of a quadrilateral are supplementary, it is a cyclic quadrilateral. 9. Surface area of a triangle = 12aha = pr =√ p(p− a)(p− b)(p− c) = abc/4R. 10. Triangle’s centroid: intersection point of medians, divides medians to 2:1. 11∗. Vector approach to geometry problems. 12. Taking derivatives: (fg)′ = fg′ + f ′g, f [g(x)]′ = f ′[g(x)]g′ (sin x)′ = cosx, (cosx)′ = − sin x (ex)′ = ex, (ln x)′ = 1/x, (xn)′ = nxn−1 (arctan x)′ = 1/(1 + x2) (arcsin x)′ = −(arccosx)′ = 1/ √ 1− x2 13. Integration: the formulas are the same as for derivatives, but with swapped left-hand-side and rhs. (inverse operation!), e.g.∫ xndx = xn+1/(n+ 1). Special case of the substitution method:∫ f(ax+ b)dx = F (ax+ b)/a. 14. Conic sections: a11x2 + 2a12xy + a22y2 + a1x + a2y + a0 = 0 with a11 = a22 — circle; with a11 ·(a11a22−a212) > 0 — ellipse, . . . < 0 — hyperbola, with a11a22 = a212 — parabola. El- lipse: l1 + l2 = 2a, α1 = α2, A = πab; hy- perbola: l1 − l2 = 2a, α1 + α2 = 0; parabola: l + h = const, α1 = α2. 15. Numerical methods. Newton’s iterative method for finding roots f(x) = 0: xn+1 = xn − f(xn)/f ′(xn). Trapezoidal rule for approximate integration:∫ b a f(x)dx ≈ b− a2n [f(x0) + 2f(x1) + . . . +2f(xn−1) + f(xn)] 16. Derivatives and integrals of vectors: differ- entiate/integrate each component; alternatively differentiate by applying the triangle rule for the difference of two infinitesimally close vectors. II General recommendations 1. Check all formulas for veracity: a) examine dimensions; b) test simple special cases (two parameters are equal, one param. tends to 0 or ∞); c) verify the plausibility of solution’s qualitative behaviour. 2. If there is an extraordinary coincidence in the problem text (e.g. two things are equal) then the key to the solution might be there. 3. Read carefully the recommendations in the problem’s text. Pay attention to the problem’s formulation — insignificant details may carry vital information. If you have solved for some time unsuccessfully, then read the text again — perhaps you misunderstood the problem. 4. Postpone long and time-consuming math- ematical calculations to the very end (when everything else is done) while writing down all the initial equations which need to be simplified. 5. If the problem seems to be hopelessly diffi- cult, it has usually a very simple solution (and a simple answer). This is valid only for Olympiad problems, which are definitely solvable. 6. In experiments a) sketch the experimental scheme even if you don’t have time for measure- ments; b) think, how to increase the precision of the results; c) write down (as a table) all your direct measurements. III Kinematics 1. For a point or for a translational motion of a rigid body (integral → area under a graph): ~v = d~xdt , ~x = ∫ ~vdt (x = ∫ vxdt etc.) ~a = d~vdt = d2~x dt2 , ~v = ∫ ~adt t = ∫ v−1x dx = ∫ a−1x dvx, x = ∫ vx ax dvx If a = Const., then previous integrals can be found easily, e.g. x = v0t+ at2/2 = (v2 − v20)/2a. 2. Rotational motion — analogous to the trans- lational one: ω = dϕ/dt, ε = dω/dt; ~a = ~τdv/dt+ ~nv2/R 3. Curvilinear motion — same as point 1, but vectors are to be replaced by linear velocities, accelerations and path lengths. 4. Motion of a rigid body. a) vA cosα = vB cosβ; ~vA, ~vB — velocities of pts. A and B; α, β — angles formed by ~vA, ~vB with line AB. b) The instantaneous center of rotation ( 6= center of curvature of material pt. trajectories!) can be found as the intersection pt. of perpendic- ulars to ~vA and ~vB , or (if ~vA, ~vB ⊥ AB) as the intersection pt. of AB with the line connecting endpoints of ~vA and ~vB . 5. Non-inertial reference frames: ~v2 = ~v0 + ~v1, ~a2 = ~a0 + ~a1 + ω2 ~R+ ~aCor Note: ~aCor ⊥ ~v1, ~ω; ~aCor = 0 if ~v1 = 0. 6∗. Ballistic problem: reachable region y ≤ v20/(2g)− gx2/2v20 . For an optimal ballistic trajectory, initial and final velocities are perpendicular. 7. For finding fastest paths, Fermat’s and Huy- gens’s principles can be used. 8. To find a vector (velocity, acceleration), it is enough to find its direction and a projection to a single (possibly inclined) axes. IV Mechanics 1. For a 2D equilibrium of a rigid body: 2 eqns. for force, 1 eq. for torque. 1 (2) eq. for force can be substituted with 1 (2) for torque. Torque is often better — “boring” forces can be eliminated by a proper choice of origin. If forces are applied only to 2 points, the (net) force application lines coincide; for 3 points, the 3 lines meet at a single point. 2. Normal force and friction force can be com- bined into a single force, applied to the contact point under angle arctanµ with respect to the normal force. 3. Newton’s 2nd law for transl. and rot. motion: ~F = m~a, ~M = I~ε ( ~M = ~r × ~F ). For 2D geometry ~M and ~ε are essentially scalars and M = Fl = Ftr, where l is the arm of a force. 4. Generalized coordinates. Let the system’s state be defined by a single parameter ξ and its time derivative ξ̇ so that the pot. energy Π = Π(ξ) and kin. en. K = µξ̇2/2; then µξ̈ = −dΠ(ξ)/dξ. (Hence for transl. motion: force is the derivative of pot. en.) 5. If the system consists of mass points mi: ~rc = ∑ mi~ri/ ∑ mj , ~P = ∑ mi~vi ~L = ∑ mi~ri × ~vi, K = ∑ miv 2 i /2 Iz = ∑ mi(x2i + y2i ) = ∫ (x2 + y2)dm. 6. In a frame where the mass center’s velocity is ~vc (index c denotes quantities rel. to the mass center): ~L = ~Lc +MΣ ~Rc × ~vc, K = Kc +MΣv2c/2 ~P = ~Pc +MΣ~vc 7. Steiner’s theorem is analogous (b — distance of the mass center from rot. axis): I = Ic+mb2. 8. With ~P and ~L from pt. 5, Newton’s 2nd law: ~FΣ = d~P/dt, ~MΣ = d~L/dt 9∗. Additionally to pt. 5, the mom. of inertia rel. to the z-axis through the mass center can be found as Iz0 = ∑ i,j mimj [(xi−xj)2 + (yi− yj)2]/2MΣ. 10. Mom. of inertia rel. to the origin θ =∑ mi~r 2 i is useful for calculating Iz of 2D bodies or bodies with central symmetry using 2θ = Ix + Iy + Iz. 11. Physical pendulum with a reduced length l̃: ω2(l) = g/(l + I/ml), ω(l) = ω(l̃ − l) = √ g/l̃, l̃ = l + I/ml 12. Coefficients for the momenta of inertia: cylinder 12 , solid sphere 2 5 , thin spherical shell 2 3 , rod 1 12 (rel. to endpoint 1 3 ), square 1 6 . 13. Often applicable conservation laws: energy (elastic bodies, no friction), momentum (no net external force; can hold only along one axis), angular momentum (no net ext. torque, e.g. the arms of ext. forces are 0 (can be written rel. to 2 or 3 pts., then substitutes conservation of lin. mom.). 14. Additional forces in non-inertial frames of ref.: inertial force −m~a, centrifugal force mω2 ~R and Coriolis force∗ 2m~v × ~Ω (better to avoid it; being ⊥ to the velocity, it does not create any work). 15. Tilted coordinates: for a motion on an inclined plane, it is often practical to align axes along and ⊥ to the plane; gravit. acceleration has then both x- and y- components. Axes may also be oblique (not ⊥ to each other), but then with ~v = vx~ex + vy~ey, vx 6= to the x-projection of ~v. 16. Collision of 2 bodies: conserved are a) net momentum, b) net angular mom., c) angular mom. of one of the bodies with respect to the impact point, d) total energy (for elastic colli- sions); in case of friction, kin. en. is conserved only along the axis ⊥ to the friction force. Also: e) if the sliding stops during the impact, the final velocities of the contact points will have equal projections to the contact plane; f) if slid- ing doesn’t stop, the momentum delivered from one body to the other forms angle arctanµ with the normal of the contact plane. 17. Every motion of a rigid body can be repre- sented as a rotation around the instantaneous center of rotation C (in terms of velocities of the body points). NB! Distance of a body point P from C 6= to the radius of curvature of the trajectory of P . 18. Tension in a string: for a massive hang- ing string, tension’s horizontal component is constant and vertical changes according to the string’s mass underneath. Pressure force (per unit length) of a string resting on a smooth surface is determined by its radius of curvature and tension: N = T/R. Analogy: surface ten- sion pressure p = 2σ/R; to derive, study the pressure force along the diameter. 19. Liquid surface takes equipot. shape (ne- glecting σ); in incompr. liquid, p = p0 − w, w—vol. dens. of pot. en. (for a gas, see X-6). 20. Bernoulli law for incompr. fluid: p+ 12ρv 2 + ρϕ = const; in homog. field, the gravit. potential ϕ = gh. For gas of specific heat cp [J/kg], 1 2v 2 + cpT = const. 21∗. Momentum continuity by straight stream- lines: p+ ρv2 = const. 22∗. Adiabatic invariant: if the relative change of the parameters of an oscillating system is small during one period, the area of the loop drawn on the phase plane (ie. in p-x coordi- nates) is conserved with a very high accuracy. 23. For studying stability use a) principle of minimum potential energy or b) principle of small virtual displacement. 24∗. Virial theorem for finite movement: a) If F ∝ |~r|, then 〈K〉 = 〈Π〉 (time averages); b) If F ∝ |~r|−2, then 2 〈K〉 = −〈Π〉. 25. Tsiolkovsky rocket equation ∆v = u ln Mm . V Oscillations and waves 1. Damped oscillator: ẍ+ 2γẋ+ ω20x = 0 (γ < ω0). Solution of this equation is (cf. I.2.): x = x0e−γt sin(t √ ω20 − γ2 − ϕ0). 2. Eq. of motion for a system of coupled oscil- lators: ẍi = ∑ j aijxj . 3. A system of N coupled oscillators has N different eigenmodes when all the oscillators oscillate with the same frequency ωi, xj = xj0 sin(ωit + ϕij), and N eigenfrequencies ωi (which can be multiple, ωi = ωj). General solu- tion (with 2N integration constants Xi and φi) is a superposition of all the eigenmotions : xj = ∑ i Xixj0 sin(ωit+ ϕij + φi) 4. If a system described with a generalized coordinate ξ (cf IV-2) and K = µξ̇2/2 has an equilibrium state at ξ = 0, for small oscilla- tions Π(ξ) ≈ κξ2/2 [where κ = Π′′(0)] so that ω2 = κ/µ. 5. The phase of a wave at pt. x, t is ϕ = kx − ωt + ϕ0, where k = 2π/λ is a wave vec- tor. The value at x, t is a0 cosϕ = <a0eiϕ. The phase velocity is vf = νλ = ω/k and group velocity vg = dω/dk. 6. For linear waves (electromagn. w., small- amplit. sound- and water w.) any pulse can be considered as a superpos. of sinusoidal waves; a standing w. is the sum of two identical counter- propagating w.: ei(kx−ωt) + ei(−kx−ωt) = 2e−iωt cos kx. 7. Speed of sound in a gas cs = √ (∂p/∂ρ)adiab = √ γp/ρ = v̄ √ γ/3. 8. Speed of sound in elastic material cs =√ E/ρ. 9. Sp. of waves in shallow (h  λ) water: v = √ gh; in a string: v = √ T/ρlin. 10. Doppler’s effect: ν = ν0 1+v‖/cs1−u‖/cs . 11. Huygens’ principle: wavefront can be con- structed step by step, placing an imaginary wave source in every point of previous wave front. Results are curves separated by distance ∆x = cs∆t, where ∆t is time step and cs is the velocity in given point. Waves travel perpendic- ular to wavefront. VI Geometrical optics. Photometry. 1. Fermat’s principle: waves path from point A to point B is such that the wave travels the least time. 2. Snell’s law: sinα1/ sinα2 = n2/n1 = v1/v2. 3. If refraction index changes continuously, then we imaginarily divide the media into lay- ers of constant n and apply Snell’s law. Light ray can travel along a layer of constant n, if the requirement of total internal reflection is marginally satisfied, n′ = n/r (where r is the curvature radius). 4. If refraction index depends only on z, the photon’s mom. px , py, and en. are conserved: kx, ky = Const., |~k|/n = Const. 5. The thin lens equation (pay attention to signs): 1/a+ 1/b = 1/f ≡ D. 6. Newton’s eq. (x1, x2 — distances of the image and the object from the focal planes): x1x2 = f2. 7. Parallax method for finding the position of an image: find such a pos. for a pencil’s tip that it wouldn’t shift with resp. to the image when moving perpendicularly the position of your eye. 8. Geometrical constructions for finding the paths of light rays through lenses: a) ray passing the lens center does not refract; b) ray ‖ to the optical axis passes through the focus; c) after refr., initially ‖ rays meet at the focal plane; d) image of a plane is a plane; these two planes meet at the plane of the lens. 9. Luminous flux Φ [unit: lumen (lm)] mea- sures the energy of light (emitted, passing a contour, etc), weighted according to the sensi- tivity of an eye. Luminous intensity [candela (cd)] is the luminous flux (emitted by a source) per solid angle: I = Φ/Ω. Illuminance [lux (lx)] is the luminous flux (falling onto a surface) per unit area: E = Φ/S. 10. Gauss theorem for luminous flux: the flux through a closed surface surrounding the point