hwk10 - homework 10 GUPTA, SHIVANI Due: Feb 23 2008, 11:00...

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homework 10 – GUPTA, SHIVANI – Due: Feb 23 2008, 11:00 pm 1 Mechanics - Basic Physical Concepts Math: Circle: 2 π r , π r 2 ; Sphere: 4 π r 2 , (4 / 3) π r 3 Quadratic Eq.: ax 2 + bx + c = 0, x = b ± b 2 4 ac 2 a Cartesian and polar coordinates: x = r cos θ, y = r sin θ , r 2 = x 2 + y 2 , tan θ = y x Trigonometry: cos α cos β + sin α sin β = cos( α - β ) sin α + sin β = 2 sin α + β 2 cos α β 2 cos α + cos β = 2 cos α + β 2 cos α β 2 sin 2 θ = 2 sin θ cos θ, cos 2 θ = cos 2 θ - sin 2 θ 1 - cos θ = 2 sin 2 θ 2 , 1 + cos θ = 2 cos 2 θ 2 Vector algebra: v A = ( A x ,A y ) = A x ˆ ı + A y ˆ Resultant: v R = v A + v B = ( A x + B x ,A y + B y ) Dot: v A · v B = AB cos θ = A x B x + A y B y + A z B z Cross product: ˆ ı × ˆ = ˆ k , ˆ × ˆ k = ˆ ı , ˆ k × ˆ ı = ˆ v C = v A × v B = v v v v v v ˆ ı ˆ ˆ k A x A y A z B x B y B z v v v v v v C = AB sin θ = A B = AB , use right hand rule Calculus: d dx x n = nx n 1 , d dx ln x = 1 x , d sin θ = cos θ , d cos θ = - sin θ , d dx const = 0 Measurements Dimensional analysis: e.g. , F = ma [ M ][ L ][ T ] 2 , or F = m v 2 r [ M ][ L ][ T ] 2 Summation: N i =1 ( ax i + b ) = a N i =1 x i + bN Motion One dimensional motion: V = ds dt , a = dv dt Average values: ¯ V = s f s i t f t i , ¯ a = v f v i t f t i One dimensional motion (constant acceleration): V ( t ) : V = V 0 + at s ( t ) : s = ¯ V t = V 0 t + 1 2 at 2 , ¯ V = v 0 + v 2 V ( s ) : V 2 = V 2 0 + 2 as Nonuniform acceleration: x = x 0 + V 0 t + 1 2 at 2 + 1 6 j t 3 + 1 24 st 4 + 1 120 k t 5 + 1 720 pt 6 + ... , (jerk, snap, ... ) Projectile motion: t rise = t fall = t trip 2 = v 0 y g h = 1 2 g t 2 fall , R = V ox t trip Circular: a c = v 2 r , V = 2 π r T , f = 1 T (Hertz=s 1 ) Curvilinear motion: a = r a 2 t + a 2 r Relative velocity: vV = vV + vu Law of Motion and applications Force: v F = mva, F g = mg, v F 12 = - v F 21 Circular motion: a c = v 2 r , V = 2 π r T = 2 π r f Friction: F static μ s N F kinetic = μ k N Equilibrium (concurrent forces): i v F i = 0 Energy Work (for all F): Δ W = W AB = W B - W A F b s = Fs cos θ = v F · vs i B A v F · dvs (in Joules) E±ects due to work done: v F ext = mva - v F c - v f nc W ext | A B = K B - K A + U B - U A + W diss | A B Kinetic energy: K B - K A = i B A mva · dvs , K = 1 2 mV 2 K (conservative v F ): U B - U A = - i B A v F · dvs U gravity = mg y , U spring = 1 2 k x 2 From U to v F : F x = - ∂ U ∂x , F y = - ∂ U ∂y , F z = - ∂ U ∂z F gravity = - ∂ U ∂y = - mg , F spring = - ∂ U ∂x = - k x Equilibrium: ∂ U ∂x = 0, 2 U ∂x 2 > 0 stable, < 0 unstable Power: P = dW dt = F V b = F V cos θ = v F · vV (Watts) Collision Impulse: v I = Δ v p = v p f - v p i i t f t i v F dt Momentum: v p = mvV Two-body: x cm = m 1 x 1 + m 2 x 2 m 1 + m 2 p cm M V cm = p 1 + p 2 = m 1 V 1 + m 2 V 2 F cm F 1 + F 2 = m 1 a 1 + m 2 a 2 =
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This note was uploaded on 10/13/2008 for the course PHY 303K taught by Professor Turner during the Spring '08 term at University of Texas at Austin.

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hwk10 - homework 10 GUPTA, SHIVANI Due: Feb 23 2008, 11:00...

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