Phy_Cheat_Sheet-print - Math Primer Sphere V= 4/3r A=4r Box...

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Math Primer Sphere V= 4/3πr³ A=4πr² Box A= 2(ab +ac +bc) Cylinder V=πr²h A=2πr² + 2πrh Chapter 2 For Constant a v² = v o ² + 2a(x-x o) Chapter 3 Projectile Motion y=y o + xtan Ө o - g x² . 2v o cos² Ө o R = (v o ²/g)(sin 2 Ө o ) H = y o + v yo ²/(2g) v yo=vert comp of vo Chapter 4 T=mg/(2n) n= #times rope over pulleys Sign: mg(l/2)sin90˚ +Mgrsin90˚ - Tlsin Ө = 0 m = post M=sign l=length of post r=d to sign = angle tween cable & post T=tension Ө Free Fall : Term Veloc v=√(mg/k) k=½C d Ap Cd=drag coef A=S.A. p=air dens Ladder : ½m l g + m m g(r/l)tan Ө = R ½m l +m m (r/l)tan Ө ≤µ(m l + m m ) ml=m ladder l=length lad mm= m man r = dist to man N=normal floor R= normal wall = tween ladder & wall Ө Chapter 5: Work, Kin Energy, Power K=½mv² 1J=1Nm=kgm²/s² 1eV = 1.602 10^(-19) J 1Cal = 4186 J 1 Mt= 4.00 10^15 J W=FΔrcosα α=90˚ =no work W=F·Δr ΔK=W Motion Up: W g = -mgh Down: W g =mgh W f = -W g W = ∫F(x)dx Springs : F= -kx W = -½kx² Power : 1W = 1J/s 1kWh = 3.6 10^6 J 1hp = 550 ft lb/s = 746 W P = dW/dt Avg P = W/(Δt) P=Fvcos(α Fv ) Chapter 6 : PE and Energy Conserv U=mgh E= K+U ΔU= -W ΔK = W Trapeze: E=mgl(1-cos ) + ½mv² Ө Springs (see chap 5): U=½kx² E=½kA² A=amplitude |v| = √[(A² - x²)(k/m)] Chapter 7 : Momentum & Collisions p=mv F = dp/dt K= p²/(2m) Impulse : J = ∫Fdt J=Δp J=F av Δt Elastic Collisions : p f,1 = (m 1 – m 2 ) p i,1 +( 2m 1 ) p i,2 (m 1 + m 2 ) (m 1 + m 2 ) p f,1 =p i,2 p f,2 =p i,1 ( special case m 1 =m 2 ) p f,1 = 2m 1 p i,2 p f,2 =(m 2 -m 1 ) p i,2 (m 1 +m 2 ) (m 1 +m 2 ) Special Case (Above) p i,1 = 0 Coeff of Restitution : ε =|v f,1 –v f,2 | ε = √(h f /h i ) h f =ε²h i |v i,1 –v i,2 | ΔK = ½ m 1 m 2 (1-ε²) (v i,1 -v i,2 (m 1 +m 2 ) Ө f = arctan( p f,┴ / p f,║ ) = arctan(ε p i,┴ / p i,║ ) < Ө i Ө f = arctan(ε tan( Ө i)) Chapter 8: Sys of Particles & Extended Objects (1/M)∑r i m i from i=1 to n M= m 1 +m 2 +…+m n Blocks : x 1 =x n +½L∑(1/i) from i=1 to n-1 n=#blocks xn=0 L=length of blocks Spherical Coordinates : x = rcosφsinυ y=rsinφsinυ z=rcosυ r = √(x² + y² + z²) υ=arccos(z/ r) φ=arctan(y/x) Cylindrical Coordinates : x=r cosφ y= r sinφ z=z r = √(x² + y²) φ= arctan (y/x) z=z (Picture goes here: Cannon Recoil : b= cannon ball cn=cannon Ө = 45˚ v b =√(gR/2) R=max range v cn =-(m b /m cn )v b Rocket Motion : v f -v i = v c ln(m i /m f ) vc=vel of propellant Chapter 9: Circular Motion Pol Coordinates : r=√(x² +y²) =arctan(y/x) Ө x=rcos y=rsin Ө Ө avg ω= Δ /Δt ω= d /dt ω=rad/s ω=2πf Ө Ө T = period (in t) of rotation f= frequency T=1/f avg α=Δω/Δt α=dω/dt s = r Ө v = rω a t = rα a c = ω²r = v²/r vec a = vec a t – veca c Chapter 10: Rotation K = ½mr²ω² K=½Iω² constant ρ I=(M/V)∫r ²dV I = ½M(R 1 ² + R 2 ²) (hollow cylinder) I = ½MR² (solid cylinder) I = MR² (cylinder thin shell) I = ¼MR² + (1/12) Mh² (solid cylinder, perpendicular) I = (1/12) Mh² (thin rod of length h, perpendicular) I = (2/5) MR² (solid sphere) I = (2/3) MR² (thin spherical shell) I = (1/12) M(a² + b²) (rectangular block) (pictures to right): 1 sidereal day= 86164 s Parallel Axis Thm : I = I cm + Md² Rolling : K = Ktrans + Krot = ½mv² + ½Iω² = (1+c)½ mv² Loops : h > ½(5+c)R N 2 nd Law: τ = Iα vecτ = vecr x vecF vecC= vecA x vecB |C|=|A||B|sin Ө Ang Momentum : vecL = vecr x vecp L = rpsin Ө dL/dt = τ Rigid Objects : L=Iω
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