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m28YuBC9feHI_1210016459_jwk572

m28YuBC9feHI_1210016459_jwk572 - midterm 04 – KIM JI –...

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Unformatted text preview: midterm 04 – KIM, JI – Due: Dec 5 2007, 7:00 pm 1 Gravity vector 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 = parenleftBig 2 π T parenrightBig 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 − ǫ cos θ r 1 = r 1+ ǫ , r 2 = r 1 − ǫ ii ) L = r m Δ r ⊥ Δ t-→ Δ A Δ t = 1 2 r Δ r ⊥ Δ t = L 2 m = const. iii ) G M a 2 = parenleftBig 2 π a T parenrightBig 2 1 a , a = r 1 + r 2 2 , T 2 = parenleftBig 4 π 2 GM parenrightBig 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 + ρg h , with P = F ⊥ A and ρ = m V F = integraltext P dA-→ ρg ℓ integraltext h ( h- y ) dy Continuity equation: Av = constant Bernoulli: P + 1 2 ρv 2 + ρg y = 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( ω t + δ ), x max = A v =- v max sin( ω t + δ ), v max = ω A a =- a max cos( ω t + δ ) =- ω 2 x , a max = ω 2 A E = K + U = K max = 1 2 m ( ω A ) 2 = U max = 1 2 k A 2 Spring: ma =- k x Simple pendulum: ma θ = mαℓ =- mg sin θ Physical pendulum: τ = I α =- mg d sin θ Torsion pendulum: τ = I α =- κθ Wave motion Traveling waves: y = f ( x- v t ), y = f ( x + v t ) In the positive x direction: y = A sin( k x- ω t- φ ) T = 1 f , ω = 2 π T , k = 2 π λ , v = ω k = λ T Along a string: v = radicalBig 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 ( ω A ) 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 π r v Spherical: Δ m Δ t = Δ m Δ V · 4 π r 2 v Sound v = radicalBig B ρ , s = s max cos( k x- ω t- φ ) Δ P =- B Δ V V =- B ∂s ∂x Δ P max = B κs max = ρv ω s max Piston: Δ m Δ t = Δ m Δ V · A Δ x Δ t = ρAv Intensity: I = P A = 1 2 ρv ( ω s max ) 2 Intensity level: β = 10log 10 I I , I = 10 − 12 W/m 2 Plane waves: ψ ( x,t ) = c sin( k x- ω t ) Circular waves: ψ ( r,t ) = c √ r sin( k r- ω t ) Spherical: ψ ( r,t ) = c r sin( k r- ω t ) Doppler effect: λ = v T , 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( k x- ωt ) + sin( k x- ω t- φ ) Standing waves: sin( k x- ω t ) + sin( k x + ω t ) Beats: sin( kx- ω 1 t ) + sin( k x- ω...
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m28YuBC9feHI_1210016459_jwk572 - midterm 04 – KIM JI –...

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