This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: homework 13 – RAMSEY, TAYLOR – Due: Nov 27 2006, 4:00 am 1 Gravity ~ F 21 = G m 1 m 2 r 2 12 ˆ r 12 , for r ≥ R , g ( r ) = G M r 2 G = 6 . 67259 × 10 11 Nm 2 /kg 2 R earth = 6370 km, M earth = 5 . 98 × 10 24 kg Circular orbit: a c = v 2 r = ω 2 r = ‡ 2 π T · 2 r = g ( r ) U = G mM r , E = U + K = GmM 2 r F = dU dr = mG M r 2 = m v 2 r Kepler’s Laws of planetary motion: i ) elliptical orbit, r = r 1 ² cos θ r 1 = r 1+ ² , r 2 = r 1 ² ii ) L = rm Δ r ⊥ Δ t→ Δ A Δ t = 1 2 r Δ r ⊥ Δ t = L 2 m = const. iii ) G M a 2 = ‡ 2 π a T · 2 1 a , a = r 1 + r 2 2 , T 2 = ‡ 4 π 2 GM · r 3 Escape kinetic energy: E = K + U ( R ) = 0 Fluid mechanics Pascal: P = F ⊥ 1 A 1 = F ⊥ 2 A 2 , 1 atm = 1 . 013 × 10 5 N/m 2 Archimedes: B = M g , Pascal=N/m 2 P = P atm + ρgh , with P = F ⊥ A and ρ = m V F = R P dA→ ρg‘ R h ( h y ) dy Continuity equation: Av = constant Bernoulli: P + 1 2 ρv 2 + ρgy = const, P ≥ Oscillation motion f = 1 T , ω = 2 π T SHM: a = d 2 x dt 2 = ω 2 x , α = d 2 θ dt 2 = ω 2 θ x = x max cos( ωt + δ ), x max = A v = v max sin( ωt + δ ), v max = ωA a = a max cos( ωt + δ ) = ω 2 x , a max = ω 2 A E = K + U = K max = 1 2 m ( ωA ) 2 = U max = 1 2 kA 2 Spring: ma = kx Simple pendulum: ma θ = mα‘ = mg sin θ Physical pendulum: τ = I α = mgd sin θ Torsion pendulum: τ = I α = κθ Wave motion Traveling waves: y = f ( x vt ), y = f ( x + vt ) In the positive x direction: y = A sin( kx ωt φ ) T = 1 f , ω = 2 π T , k = 2 π λ , v = ω k = λ T Along a string: v = q F μ Reflection of wave: fixed end: phase inversion open end: same phase General: Δ E = Δ K +Δ U = Δ K max P = Δ E Δ t = 1 2 Δ m Δ t ( ωA ) 2 Waves: Δ m Δ t = Δ m Δ x · Δ x Δ t = Δ m Δ x · v P = 1 2 μv ( ωA ) 2 , with μ = Δ m Δ x Circular: Δ m Δ t = Δ m Δ A · Δ A Δ r · Δ r dt = Δ m Δ A · 2 πrv Spherical: Δ m Δ t = Δ m Δ V · 4 πr 2 v Sound v = q B ρ , s = s max cos( kx ωt φ ) Δ P = B Δ V V = B ∂s ∂x Δ P max = B κs max = ρvωs max Piston: Δ m Δ t = Δ m Δ V · A Δ x Δ t = ρAv Intensity: I = P A = 1 2 ρv ( ωs max ) 2 Intensity level: β = 10log 10 I I , I = 10 12 W/m 2 Plane waves: ψ ( x,t ) = c sin( kx ωt ) Circular waves: ψ ( r,t ) = c √ r sin( kr ωt ) Spherical: ψ ( r,t ) = c r sin( kr ωt ) Doppler effect: λ = vT , f = 1 T , f = v λ Here v = v sound ± v observer , is wave speed relative to moving observer and λ = ( v sound ± v source ) /f , detected wave length established by moving source of frequency f . f received = f reflected Shock waves: Mach Number= v source v sound = 1 sin θ Superposition of waves Phase difference: sin( kx ωt )+sin( kx ωt φ ) Standing waves: sin( kx ωt )+sin( kx + ωt ) Beats: sin( kx ω 1 t )+sin( kx ω 2 t ) Fundamental modes: Sketch wave patterns String: λ 2 = ‘ , Rod clamped middle: λ 2 = ‘ , Openopen pipe: λ 2 = ‘ , Openclosed pipe: λ 4 = ‘ Temperature and heat...
View
Full
Document
This homework help was uploaded on 04/06/2008 for the course PHY 303K taught by Professor Turner during the Spring '08 term at University of Texas at Austin.
 Spring '08
 Turner
 Physics, Gravity, Work

Click to edit the document details