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Homework 13

# Homework 13 - homework 13 – BAUTISTA ALDO – Due 4:00 am...

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Unformatted text preview: homework 13 – BAUTISTA, ALDO – 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 = ‘ , Open-open pipe: λ 2 = ‘ , Open-closed pipe: λ 4 = ‘ Temperature and heat...
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Homework 13 - homework 13 – BAUTISTA ALDO – Due 4:00 am...

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