Daniel Guetta (guetta@cantab.net) Massachusetts Institute of Technology Department of Physics 8.01 ~ FINAL EXAM EQUATION SUMMARY 1. COMMON INTERACTIONS 2. SYSTEMS OF PARTICLES & RIGID BODIES cmtotal total1 1di iim mm m= =R r r cmtotal total1 1di iim mm m= =V v v Position vector of CoM Mass of particle i Position vector of particle i Total mass of system Velocity of CoM Velocity of particle i Integrate over rigid body Moment of Inertia (MoI) 2 2, ,di i iiI m r r m^ ^= =Sum over all particles in system Perpendicular distance of particle i to the axis MoI about a particular axis Parallel Axis Theorem 2D CMI I mD = +CMIDIDMoI about an axis parallel to the axis through the CoM and a distance D away from it MoI about an axis through the CoM Mass of rigid body Centre of Mass (CoM) Sum over all particles in system 1 21,2 1,21,2m mGr= - F r1m2m1,2r1,2 r(attractive) Near the surface of the earth, |F| = mg, directed towards the earths surface. Friction 0s sf N m Static friction: Kinetic friction: k kf N m = opposes motion. Hookes Law k = - F x(restoring) xnatural length = r F t Torque FOrigin rUniversal Law of Gravity Daniel Guetta (guetta@cantab.net) 3. KINEMATICS 4. SIMPLE HARMONIC MOTION 5. CENTRAL FORCE MOTION Definitions Period2 / T p w = Frequency1/ f T =Motion ( ) cos( ) sin( ) x t A t B t w w = +Two-body problem Elliptical orbits q rTwobodiesofmassesm1andm2behavelikeasingle body of mass orbiting around the common centre of mass, where 1 21 2m mm mm =+q22 222 2eff eff1( )21 1( )2 21( )2 2E mv U rmv mv U rLmv U rrK Um= += + += + += +rrqDefinitions 22d d dd d d t t t= = =x v xv a Linear: 22d d dd d d t t tq w qw a = = = Rotational: Circular Motion RqRqRwq rArc length:s Rq =Velocityd dR Rtqw = v q = qAcceleration: 2 222d dvR R RR tqw a = - + = - + a r r q qDaniel Guetta (guetta@cantab.net) 6. WORK & ENERGY 7. DYNAMICS cmvwrForabodytranslating androtatingaboutits centre of mass Origin For a point particle, only the translational part need be considered. Forabodyrotatingbutnottranslating,onlythe rotationalpartneedbeconsidered,butthemoment of inertia I must be taken about the correct axis. Linear Momentum cm cmTranslationalRotationalmotionmotionm I = + L r v w mm vm = p v Linear Impulse end timestart timedt =I F Angular Impulse end time end timestart time start timed d t t = = = J r F r I t Impulse/Momentum Principle after impulse before impulse= - = D I p p p after impulse before impulse= - = D J L L L ddIt=Lt = a ddmt= =pF aAngular Momentum Equations of Motion Potential Energy Functions Spring: 212PE k = xKinetic Energy 2 12mv Linear: 2 12I w Rotational: Work-Energy Theorem initial initial final finalU KE U KE W + = + +xnatural length (With PE = 0 at equilibrium) Universal gravitation: 1m2mr1 2m mPE Gr= -(With PE = 0 at r = ) Gravity near the earths surface: mh PE mgh = -(With PE = 0 at the earths surface) Work done by a Force final positionoriginal positiond W= F rPotentials & Conservative Forces at position of 0 potentialdrrPE = - F r