{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

final_macdonald

# final_macdonald - nal 01 KELLERMANN MARC Due 11:00 pm...

This preview shows pages 1–2. Sign up to view the full content.

final 01 – KELLERMANN, MARC – Due: Dec 19 2006, 11:00 pm 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 N m 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 m M r , E = U + K = - G m M 2 r F = - d U dr = - m G M r 2 = - m v 2 r Kepler’s Laws of planetary motion: i ) elliptical orbit, r = r 0 1 - cos θ r 1 = r 0 1+ , r 2 = r 0 1 - 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 = 2 π a T 2 1 a , a = r 1 + r 2 2 , T 2 = 4 π 2 G M 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 + ρ g h , with P = F A and ρ = m V F = P dA -→ ρ g h 0 ( h - y ) dy Continuity equation: A v = constant Bernoulli: P + 1 2 ρ v 2 + ρ g y = const, P 0 Oscillation motion f = 1 T , ω = 2 π T S H M: 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: m a = - k x Simple pendulum: m a θ = m α = - m g sin θ Physical pendulum: τ = I α = - m g d sin θ Torsion pendulum: τ = I α = - κ θ 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 = 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 ( ω A ) 2 Waves: Δ m Δ t = Δ m Δ x · Δ x Δ t = Δ m Δ x · v P = 1 2 μ v ( 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 = B ρ , s = s max cos( k x - ω t - φ ) Δ P = - B Δ V V = - B s x Δ P max = B κ s max = ρ v ω s max Piston: Δ m Δ t = Δ m Δ V · A Δ x Δ t = ρ A v Intensity: I = P A = 1 2 ρ v ( ω s max ) 2 Intensity level: β = 10 log 10 I I 0 , I 0 = 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 e ff ect: λ = v T , f 0 = 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 0 , detected wave length established by moving source of frequency f 0 . f received = f reflected Shock waves: Mach Number= v source v sound = 1 sin θ Superposition of waves Phase di ff erence: sin( k x - ω t ) + 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 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 Conversions: F = 9 5 C + 32 , K = C + 273 . 15 Constant volume gas thermometer: T = a P + b Thermal expansion: α = 1 d dT , β = 1 V d V dT Δ = α Δ T , Δ A = 2 α A Δ T , Δ V = 3 α V Δ T Ideal gas law: P V = n R T = N k T R = 8 . 314510 J / mol / K = 0 . 0821 L atm / mol / K k = 1 . 38 × 10 - 23 J / K, N A = 6 . 02 × 10 23 , 1 cal=4.19 J Calorimetry: Δ Q = c m Δ T, Δ Q = L Δ m First law: Δ U = Δ Q - Δ W , W = P dV Conduction: H = Δ Q Δ t = - k A Δ T Δ , Δ T i = - H A i k i Stefan’s law: P = σ A e T 4 , σ = 5 . 67 × 10 - 8 W m 2 K 4 Kinetic theory of gas Ideal gas: Δ p x = 2 m v x , F = Δ p x Δ t

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 16

final_macdonald - nal 01 KELLERMANN MARC Due 11:00 pm...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online