Concourse 8.01 Formula Sheet December 1, 2005Kinematics∆~r = ~r −~r0~vavg=∆~r∆t~aavg=∆~v∆t~v =d~rdt~a =d~vdt=d2~rdt2average speed =distance travelled∆tin Cartesian components (3-D): in plane polar components (2-D):~r = xˆı + yˆ + zˆk ~r = rˆr~v = ˙xˆı + ˙yˆ + ˙zˆk ~v = ˙rˆr + r˙θˆθ~a = ¨xˆı + ¨yˆ + ¨zˆk ~a =¨r − r˙θ2ˆr +r¨θ + 2 ˙r˙θˆθAngular Kinematics∆θ = θ − θ0ωavg=∆θ∆tαavg=∆ω∆tω =dθdtα =dωdt=d2θdt21-D motion (x or θ) with constant acceleration (a or α)v = v0+ at ω = ω0+ αtx = x0+ v0t +12at2θ = θ0+ ω0t +12αt2v2− v20= 2a(x − x0) ω2− ω20= 2α(θ − θ0)vavg=v + v02ωavg=ω + ω02Uniform Circular Motion~v = Rωˆθ~a = −Rω2ˆr = −v2RˆrNon-uniform Circular Motion~v = Rωˆθ~a = −Rω2ˆr + R αˆθRelative Velocity~v(C relative to A)= ~v(C relative to B)+ ~v(B relative to A)~v(B relative to A)= −~v(A relative to B)page 1 of 6Concourse 8.01 Formula Sheet December 1, 2005Simple Harmonic MotionThe differential equationd2x(t)dt2+ ω02x(t) = 0represents a simple harmonic oscillator, and has solutions of the formx(t) = A cos(ω0t + φ)where:ω0is the angular frequency.A and φ are arbitrary constants that depend on the initial conditions.f =1T=ω02π(relation between frequency, p eriod, and angular frequency)ω0=rkm(angular frequency of a simple m ass -spring oscillator)Rolling Without SlippingVcm= ±RωDynamicsX~F = m~a (Newton’s 2nd Law for a single particle)FG=Gm1m2r2(gravitational force of attraction betwee n two particles)FG= mg (gravitational force on a m ass m near the surface of the Earth)f = µkN (kinetic friction)f ≤ µsN (static friction)F = −kx (Hooke’s Law: linear restoring force)Momentum~p = m~v (momentum of a particle)X~F =d~pdt(Newton’s 2nd Law for a single particle, in terms of momentum)~Favg=∆~p∆t(defintion of average [net] force)page 2 of 6Concourse 8.01 Formula Sheet December 1, 2005Systems of ParticlesM =Xjmj(total mass of a system of particles)~P =Xj~pj(momentum of a system of particles)X~Fext=d~Pdt(Newton’s 2nd Law for a system of particles)Center of Mass~Rcm=Pjmj~rjPjmj(center of mass of a system of particles)~P = M~Vcm= Md~Rcmdt(momentum of a system of particles, in terms of CM)X~Fext= M~Acm= Md2~Rcmdt2(Newton’s 2nd Law for a system of particles, in terms of CM)Thrust EquationMd~vdt=X~Fext+ ~ureldMdt~urelis the velocity of the e ntering (or leaving) mass particles relative to the main object.Rotational Dynamics~L = ~r × ~p (angular momentum of a particle about the origin)~τ = ~r ×~F (torque about the origin due to a force~F )X~τ =d~LdtFor a system of particles,X~τext=d~LdtRigid BodiesI =XjmjR2j(moment of inertia of a rigid body)Ik= Icm+ Md2(Parallel Axis Theorem)Iz= Ix+ Iy(Perpendicular Axis Theorem, for a 2-D object in the xy plane)page 3 of 6Concourse 8.01 Formula Sheet December 1, 2005For a rigid body rotating about the z axis,Lz= IzωXτz= IzαKnown Moments of InertiaI = M R2(thin ring)I =112ML2(uniform thin ro d, about axis through center)I =12MR2(uniform disc or cylinder)I =25MR2(uniform solid sphere)Simultaneous Rotation and TranslationFor a rigid body which is both translating and rotating (about an axis through its center of mass, oriented alongthe z direction),Lz= (~Rcm× M~Vcm)z+ IzωXτz= Izα about CMWork and EnergyXW = ∆K (Work-Energy Theorem)W =Z~rb~ra~F · d~r (work done by~F as particle moves from ~rato ~rb)K =12mv2(kinetic energy of a particle)P wr =dWdt=~F ·~v (power)For a rigid body rotating about the z axis,K =12Izω2For a rigid body which is both translating and rotating (about an axis through its center of mass, orientedalong the z direction),K =12MV2cm+12Iω2about cmpage 4 of 6Concourse 8.01 Formula Sheet December 1, 2005Potential Energy∆U = −Z~rb~ra~F · d~r (difference in potential energy due to a conservative force~F )U(~r) = U (~r0) −Z~r~r0~F · d~r (potential energy defined with respect to a reference point)Fx= −dUdx(finding force from U, in one dimension)E = K + U (total mechanical energy)XWNC= ∆E (Work-Energy Theorem restated in terms of total mechanical energy)U =−Gm1m2r(potential energy due to gravity, universal law)U = mgy (potential energy due to gravity, near Earth’s surface)U =12kx2(potential energy due to spring force)Math Corner“Dot” Notation˙f ≡dfdtVectors~A ·~B =~A~Bcos θA,B= ABk to A= Ak to BB~A ×~B=~A~Bsin θA,B= AB⊥ to A= A⊥ to BB... in Cartesian components~A = Axˆı + Ayˆ + Azˆk~A=qAx2+ Ay2+ Az2~A +~B = (Ax+ Bx)ˆı + (Ay+ By)ˆ + (Az+ Bz)ˆk~A ·~B = AxBx+ AyBy+ AzBz~A ×~B =ˆı ˆˆkAxAyAzBxByBz= ˆı (AyBz− AzBy) − ˆ (AxBz− AzBx) +ˆk (AxBy− AyBx)Coordinate Conversionx = r cos θ r =px2+ y2ˆr = ˆı cos θ + ˆ sin θy = r sin θ θ = arctanyxˆθ = −ˆı sin θ + ˆ cos θpage 5 of 6Concourse 8.01 Formula Sheet December 1, 2005GeometryA sphere of radius R has volume43πR3and surface area 4πR2.A cylinder of radius R and height h has volume πR2h and surface area 2πRh + 2πR2(the first term is the areaaround the side, the second term is the area of the top and bottom).Trigonometrysin2θ + cos2θ = 1 sin2θ =12−12cos(2θ) sin 45◦= cos 45◦=1√2sin(θ + φ) = sin θ cos φ + cos θ sin φ cos2θ =12+12cos(2θ) sin 30◦= cos 60◦=12cos(θ + φ) = cos θ cos φ − sin θ sin φ sin(2θ) = 2 sin θ cos θ sin 60◦= cos 30◦=√32Quadratic FormulaIf ax2+ bx + c = 0 then x =−b ±√b2− 4ac2a.page 6 of
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