Physics Problems Tutorial Materials

Wgrav ! )m g " y (work done by gravity) Physics 2325 - Formula List ! 1D Kinematics General 1D Motion: x as a function of t "x vave ! "t . v! #x #t , aave ! "v . a! "t F ! )k x (Hooke's Law), W ! ) k x2f ) xi2  2 1 #v ! Work-Energy Theorem Wnet ! " K #t Constant Acc.: v$t% ! v0 & a t and x$t% ! x0 & v0 t & " x ! v0 t & 1 2 at 2 xf ! Springs: Hooke's Law and Work Constant Vel.: x$t% ! x0 & v t ' " x ! v t v ! v0 & a t W ! 9 xi F $x% # x (variable force in 1D) "x! 1 $v0 & v% t 2 v2 ! v20 & 2 a " x 1 2 at 2 * ! General Vectors A ! Ax x &Ay y ! Ax , Ay  Mag. & Dir. angle " Components Ax ! A cos Θ and Ay ! A sin Θ . Components " Mag. & Dir. angle A! A2x & A2y and Θ ! Ay tan)1 Ax Ay 180 / & tan)1 Ax ! General 2D Kinematics  "r vave ! "t , v!  #r #t , aave ! for Ax . 0 "t #v , a! #t , Ave. Speed ! Total Distance Total Time 2 Horizontal: ax ! 0 ' vx is const. v0x ! v0 cos Θ and v0y ! v0 sin Θ. Vertical: ay ! )g 2 g Second Law: F net ! m a Third Law: F 12 ! )F 21 ! Friction between surfaces fs 4 Μs N (static), fk ! Μk N (kinetic) ! Circular Motion v2 2Πr 2Π 2 , Also v ! ' ac !   r r T T #v #t ! Accelerated Frames gart ! )a (artificial gravity) accelerated frame ' false force opposite acc. ! Dot or Scalar Product A # B ! A B cos Θ where A ! A # B ! A x B x & A y B y & Az Bz ! Work W ! 9 F # # r Power: > ! #W #t ! F ?v # #x 3D: Fx ! ) U, @ @x U , Fy ! ) @ @y U , Fz ! ) ! Momentum and Impulse-Momentum Theorem p ! m v (mom.)   @ @z U I ! 9 ti F # t (impulse) tf F net,ave " t ! I net ! " p ! m v f ) vi  ! Center of Mass for a System of Particles rcm ! 1  Ci mi ri M 1  rcm ! 9 r #m M M ! Ci mi Discrete : Continuous : M ! 9 # m F net ! m a F net ! ! Conservation of Momentum Weight 3 Mass: W ! m g r ' Ebottom ! Etop ' v2bottom ! v2top & 2 g h # #t # #t ptot ! m1 v1 &m2 v2 & ... ! M vcm First Law: v is const., unless net force. , at ! Wnc ! 0 , one mass, gravity is only cons. force ext system : F ext net ! M acm F net ! ! Newton's Laws General Circular Motion: ac ! E ! Emech ! Ktot & Utot ' Ei & Wnc ! E f , Wnc! 0 ' Ei ! E f particle : ! Relative Motion v ! v2 & v0 v2 1 k x2 2 ! Second Law for a Particle and System v R ! 0 sin$2 Θ%, (R ! " x when " y ! 0) Uniform Circular Motion: ac ! Spring: U ! 1D: F ! ) 1 Constant Acc.: v$t% ! v0 & a t and r$t% ! r0 & v0 t & a t2 ! Projectiles Gravity: U ! m g y F is conservative < 0 ! = F # # r conservative forces ' " U ! )W (U is potential energy) ! Potential Energy " Force for Ax 0 0 r as a function of t "v 1 m v2 (kinetic energy) 2 Wnc is work of all nonconservative forces. Free Fall : x ( y $y is up% and a ! )g * Wnet (net work). K ! A2x & A2y & A2z const. force: W ! F " x (1D) W ! F # " r (2D or 3D) p ptot F ext net ! 0 ' " ptot ! 0 ' ptot,i ! ptot, f ext ! 0 ' " p Fnet,x tot,x ! 0 ' ptot,i,x ! ptot, f ,x ! Collisions ptot,i ! ptot, f ' m1 v1i & m2 v2i ! m1 v1 f & m2 v2 f Elastic < Ktot,i ! Ktot, f 1D Elastic trick: Ktot,i ! Ktot, f ' v1i &v1 f ! v2i &v2 f Totally Inelastic: v1 f ! v2 f ! v f ' m1 v1i & m2 v2i ! $m1 & m2 % v f ! General Rotations about fixed axis: Ωave ! "Θ "t , Ω! #Θ #t , Αave ! "Ω "t ! Constant Angular Acceleration Ω ! Ω0 & Α t "Θ ! Ω0 t & 1 Α t2 2 "Θ ! , 1 $Ω & Ω0 % t 2 Ω2 ! Ω20 & 2 Α "Θ Α! #Ω #t ! Rotational and Linear Quantities * ! Gravitational Potential Energy v ! r Ω ut or vt ! r Ω , vc ! 0 * * Two masses: U ! )G a ! r Α ut &Ω2 r uc or at ! r Α , ac ! Ω2 r Α ! 0 < Ω ! const ! ! Moment of Inertia Escape speed: vesc ! 2Π T a I b rectangular plate about F axis thru. center: I ! Sphere about axis thru. center: 2 thin shelled hollow: I ! M R2 , 3 1 M a2 & b2  12 2 solid: I ! M R2 5 Hoop about F axis thru. center: I ! M R2 ! Rotational Energy K! * * ! Cross or Vector Product A $ B ! u A B sin Θ , right hand rule J u * * * x y z * A y Az * A x Az * A x A y A $ B ! A x A y Az ! x )y &z B y Bz B x Bz Bx By B x B y Bz ! Torque About Origin: Τ ! r L F, About Axis: Τ ! r FF ! rF F ! r F sin Θ Torque due to gravity: Τgravity ! rcm L M g ! Angular Momentum of Particle About Origin: L ! r L p, About Axis: L ! r pF ! rF p ! r p sin Θ ! General Rigid Body Dynamics 2nd Law: Τnet ! I Α and Τnet ! Angular Momentum: L ! I Ω ! System of Particles Τext net ! #L #t # Ltot #t Τext net ! 0 ' " Ltot ! 0 (Conservation) ! Equilibrium F net ! 0 and Τnet ! 0 ! Newton's Law of Gravity Magnitude: F ! G m1 m2 r2 Direction is attractive. m * Discrete Distribution: F ! )G m  2i ri ri * Continuous Distribution: F ! )G m  2 # m r r Sph. Shell: F ! G g!G M R2 Mm r2 $r . R%, F ! 0 $r 0 R% (at surface of spherical planet) #2 x ! )Ω2 # t2 4 Π2 3 r GM x, Ω ! 2 Π f ! v ! QΩ Ω 2Π T 1 k A2 1 1 m v2 & k x2 !  2 (mass/spring) 1 2 2 m v2max 2 A2 ) x2 and vmax ! Ω A (in general) ! Examples of Simple Harmonic Motion kRm Physical Pendulum: Ω ! 1 M R2 (same as solid cylinder) 2 1 1 1 I Ω2 , U ! M g ycm , Ktot ! Icm Ω2 & M v2cm 2 2 2 ! Energy E ! Mass/Spring: Ω ! (same as thin-shelled hollow cylinder) Disk about F axis thru. center: I ! and T 2 ! v x $t% ! A cos $Ω t &Φ% ! x0 cos Ω t & 0 sin Ω t Moments for uniform bodies: Thin rod about F axis 1 1 M L2 , thru. center: I ! M L2 3 12 M r ! Simple Harmonic Motion Continuous: I ! 9 r2 # m Parallel-axis Theorem: I ! Icm & H d2 thru. end: I ! 2GM R ! Circular Orbits v2 ! G I for a distribution - r is F dist. from axis Discrete: I !  mi ri2 , mi m j Mm , Several masses: U ! )G C i0 j r ri j Simple Pendulum: Ω ! m g dRI gRL @2 1 @2 u! 2 2 u v @t General Solution: u $x, t% ! f $x )v t% &g $x &v t% ! 1D Wave Equation @ x2 ! Sinusoidal Waves u $x, t% ! A cos $k x Q Ω t &Φ% Λ! 2Π , k f! Ω 2Π , v! f Λ! Ω k ! Waves on a String u $x, t% J y$x, t% Speed: v ! FT R Μ where FT ! Tension, Power: > ! ! Temperature Scales TF ! 1 Μ Ω2 A2 v 2 9 9 TC &32, "T F ! "T C , TK ! TC & 273 5 5 ! Heat Q ! Heat added to system Q ! m c " T (Temp. change), Q ! Q m L (phase change) ! Ideal Gas Law P V ! N k T and P V ! n R T n ! # of moles, N ! NA n ! # of molecules Masses: mtot ! n mmole ! N mmolecule ! Work W ! 9 P # V ! Q Area (done by system) Constant P: W ! P " V , Ideal gas at constant T: W ! n R T ln $Vf R Vi % ! First Law " U ! Q )W, # U ! ∆ Q )∆W, $∆ is inexact differential% ! Entropy #S ! Const. T: " S ! Q T ∆Q T ' "S !  ∆Q T T , Changing T: " S ! m c ln f Ti ! Second Law For a thermally isolated system: "Stot U 0 ! Heat Engines QH ! QC & W, Efficiency: e ! Q W !1) C QH QH T Max. Eff.: emax ! 1 ) C (Carnot Engine is H.E. of max. eff.) TH