Formulae

Abstract

Vectors: A = Axi + Ayj + Azk Scalar Product (“Dot” Product) A·B = AxBx + AyBy + AzBz = AB cosθA,B= AB// = A//B Vector Product (“Cross” Product) direction: “right-hand rule” A×B = (AyBz–AzBy)i +(AzBx–AxBz)j +(AxBy–AyBx)k = |A×B| = AB sinθA,B= AB⊥ = A⊥B Kinematics: the “motion”: position s as function of time t s = s(t) (position) velocity v; acceleration a: v ≡ ds/dt (speed ≡ |v|); a ≡ dv/dt Linear motion with constant a: v = v0 + at, s = s0 +v0t + 1⁄2at; eliminating t: v = v0 + 2a·(s – s0) rotation angle θ (radians; rotation radius R): θ ≡ s(=arc length)/R = θ(t) (angular position) angular velocity ω ; angular acceleration α : ω ≡ vT/R ; α ≡ aT/R (T = Tangential) Circular motion (radius R) with constant α: ω = ω0 + αt; θ = θ0 +ω0t + 1⁄2αt; eliminating t: ω = ω0 + 2α(θ –θ0) circular motion – radial acceleration arad: arad = ac = vT/R (radially inwards) Center-of-Mass position of system (mass M) rcm≡Σjmjrj/Σjmj=Σjmjrj/M=∫rdm/M, vcm=Σjmjvj/M Moment of Inertia I: I ≡ Σmiri = ∫rdm; ri(r) = distance between rotation axis and mi(dm) thin uniform rod (M, L), axis ⊥ rod through cm: I = ML/12 hollow cylinder (M, Rin, Rout), axis is cylinder axis: I = 1⁄2 M(Rin+Rout) uniform solid sphere (M, R), axis through center: I = 2MR/5 Parallel-Axis Theorem I = I//,cm + Md (d=distance between the parallel axes) Perpendicular-Axis Theorem for a Planar (=flat) body in the x-y plane: Iz = Ix + Iy Momentum p p ≡ Σimivi = Mvcm Angular Momentum L L ≡ ΣjRj×mjvj (= Iω for rotation around symmetry axis) Moving axis: Ktot ≡ 1⁄2 mvcm + 1⁄2 Icmωcm Kinetic Energy K [J≡Nm]: Fixed axis A: Krot,A ≡ 1⁄2 IAωA Forces [N≡kgm/s] and consequences: ΣjFj = dp/dt = ma; FA on B = –FB on A ; Σjτj = dL/dt = Iα Force of Gravity between M and m, at center-to-center distance r FG = GMm/r (–r) (attractive! G=6.67×10 Nm/kg); near sea level: FG = mg(–j) (downwards; g=9.80 m/s) Force of a Spring (spring constant k): FS = –kx (opposes compression/stretch x) Friction: static: Ff ≤ μsN, kinetic: Ff = μkN, opposes motion; μ=frict’n coef.; N=normal force Torque: τ ≡ R×F (τ = RFsinθR,F; direction: right-hand rule) Equilibrium & Collisions: if ΣjFj = dptot/dt = 0 ⇒ Δptot = 0; if Σjτj = dLtot/dt = 0 ⇒ ΔLtot = 0 Elastic: K is conserved; Completely Inelastic: objects stick together afterwards Impulse by a force F over a time interval: JF ≡ ∫Fdt Work done by a force F over a trajectory: WF ≡ ∫F⋅dx Work done by a torque τF over a rotation angle θ: WF ≡ ∫τF ⋅dθ Work-Kinetic Energy relationship (from ΣFj= ma): Wtot = ΣjWj = ΔK ≡ Kf – Ki Power P [W≡J/s]: PF ≡ dWF/dt = F·v = τF·ω Potential Energy U of a conservative force F: UF = –WF ; e.g. UG = –GMm/r, US = 1⁄2kx Work by Non-Conservative forces Total Energy: WNC.= ΔE = Ef –Ei ≡ Kf +ΣjUfj – (Ki+ΣjUij) x A y z A A