Kleinmans Final - final 01 KELLERMANN, MARC Due: Dec 16...

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

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: final 01 KELLERMANN, MARC Due: Dec 16 2006, noon 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 = , Open-open pipe: 2 = , Open-closed pipe: 4 = Temperature and heat...
View Full Document

Page1 / 14

Kleinmans Final - final 01 KELLERMANN, MARC Due: Dec 16...

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

View Full Document Right Arrow Icon
Ask a homework question - tutors are online