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Unformatted text preview: homework 11 RAMSEY, TAYLOR Due: Nov 13 2006, 4:00 am 1 Gravity ~ F 21 = G m 1 m 2 r 2 12 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 = 2 r = 2 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 Keplers Laws of planetary motion: i ) elliptical orbit, r = r 1 cos r 1 = r 1+ , r 2 = r 1 ii ) L = rm r t A t = 1 2 r r t = L 2 m = const. iii ) G M a 2 = 2 a T 2 1 a , a = r 1 + r 2 2 , T 2 = 4 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 + gh , with P = F A and = m V F = R P dA g R h ( h y ) dy Continuity equation: Av = constant Bernoulli: P + 1 2 v 2 + gy = const, P Oscillation motion f = 1 T , = 2 T SHM: a = d 2 x dt 2 = 2 x , = d 2 dt 2 = 2 x = x max cos( t + ), x max = A v = v max sin( t + ), v max = A a = a max cos( t + ) = 2 x , a max = 2 A E = K + U = K max = 1 2 m ( A ) 2 = U max = 1 2 kA 2 Spring: ma = kx Simple pendulum: ma = m = mg sin Physical pendulum: = I = mgd sin Torsion pendulum: = I = Wave motion Traveling waves: y = f ( x vt ), y = f ( x + vt ) In the positive x direction: y = A sin( kx t ) T = 1 f , = 2 T , k = 2 , v = k = T Along a string: v = q F Reflection of wave: fixed end: phase inversion open end: same phase General: E = K + U = K max P = E t = 1 2 m t ( A ) 2 Waves: m t = m x x t = m x v P = 1 2 v ( A ) 2 , with = m x Circular: m t = m A A r r dt = m A 2 rv Spherical: m t = m V 4 r 2 v Sound v = q B , s = s max cos( kx t ) P = B V V = B s x P max = B s max = vs max Piston: m t = m V A x t = Av Intensity: I = P A = 1 2 v ( s max ) 2 Intensity level: = 10log 10 I I , I = 10 12 W/m 2 Plane waves: ( x,t ) = c sin( kx t ) Circular waves: ( r,t ) = c r sin( kr t ) Spherical: ( r,t ) = c r sin( kr t ) Doppler effect: = vT , f = 1 T , f = v Here v = v sound v observer , is wave speed relative to moving observer and = ( 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 Superposition of waves Phase difference: sin( kx t )+sin( kx t ) Standing waves: sin( kx t )+sin( kx + t ) Beats: sin( kx 1 t )+sin( kx 2 t ) Fundamental modes: Sketch wave patterns String: 2 = , Rod clamped middle: 2 = , Openopen pipe: 2 = , Openclosed pipe: 4 = Temperature and heat...
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 Spring '08
 Turner
 Physics, Gravity, Work

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