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! SOCT (Student Opinion of Courses and Teaching) Online Forms: PHY 183 4/17/2006 5/12/2006 http://rateyourclass.msu.edu ! Final Exam date: Thurs. May 4, 810 pm E100 VMC (Veterinary Medical Center) Bldg. ! Clicker points (thru April 18) posted on LonCapa Maximum points so far: 146 Scores out of ! maximum = 73 Physics for Scientists & Engineers 1
Spring Semester 2006
Lecture 51 April 21, 2006 Physics for Scientists&Engineers 1 1 April 21, 2006 Physics for Scientists&Engineers 1 2 ! Suppose member of chain at x=0 oscillates with Mathematical Description of Waves
y ( x = 0, t ) = A sin(!t + "0 ) Wave Number and wave function
! Defining the wave number: similar to " = ! Note that v = != ! Time delay for the oscillation to move distance x is 2! T
" 2# /T = T 2# / "
$ 2" # x !t = v x y ( x, t ) ! y (0, t " ) v v= ! " ! Then we can write for the wave y (x,t): ! The wave function reads as: ! y(x,t) = Asin (! t " # x + $0 )
y ( x, t + T ) = y ( x, t ) y ( x + ! , t ) = y ( x, t )
Physics for Scientists&Engineers 1 4 x # # x$ $ y ( x, t ) = y (0, t % ) = A sin & ! & t % ' + "0 ' v ( ( v) ) ! One can check that: April 21, 2006 Physics for Scientists&Engineers 1 3 April 21, 2006 Quiz 51.1
! A transverse wave on a string is described by the formula Waves in 2d
! Now we have an oscillation in the zdirection, as a function of time and x and ycoordinate y(x,t) = 0.83 sin(8t " 2x)
where x is in meters and t is in seconds. ! Select the correct choice for the direction and speed of the wave. ! A) Direction +x, Speed 8 m/s B) Direction x, Speed .25 m/s C) Direction x, Speed 2 m/s D) Direction +x, Speed 2 m/s E) Direction +x, Speed 4 m/s Plane wave Z(x,y,t) = A sin(!x"t) t=0 Circular wave Z(x,y,t) = A sin(!r"t)/r1/2, r = (x2+y2)1/2, t = 0 April 21, 2006 Physics for Scientists&Engineers 1 5 April 21, 2006 Physics for Scientists&Engineers 1 6 Coupled Oscillators  Quantitative
Movement restricted to up/down by (blue) guide wires Coupled to neighbors with springs y Equation for Coupled Oscillators
! Equation of motion for mass n (F=ma):
man = F+ + F! = !k ( yn ! yn +1 ) ! k ( yn ! yn !1 ) m "2 y n = k ( yn +1 ! 2 yn + yn !1 ) "t 2
1st derivative: !x x yn !1
Equilibrium positions F+ = !k(yn +1 ! yn ) yn +1 yn ! Approximation "y #y yn +1 ! yn $ = "x #x #x 2 nd '2 y derivative: = 2 * 'x # "y $ " % & ( yn +1 ! yn ) ! ( yn ! yn !1 ) y ! 2 yn + yn !1 ( "x ) = "x "x = n +1 "x "x "x 2 yn = y ( xn , t )
April 21, 2006 F! = !k(yn ! yn !1 ) k = spring constant
Physics for Scientists&Engineers 1 7 April 21, 2006 !2 y !2 y 2 " m 2 # k 2 $x !t !x
Physics for Scientists&Engineers 1 8 General Wave Equation
! Wave equation applicable for all wave motion in 1D: ! Musical sound produced by inducing vibration on strings ! Decompose string into a series oscillators separated by ! The velocity of wave is Wave on a string "2 "2 y(x,t) # v 2 2 y(x,t) = 0 "t 2 "x
! Solutions: !2 y !2 y 2 " m 2 # k 2 $x !t !x !x v= ! y1 (x,t) = Y (t " x /v) with Y = arbitrary function! y 2 (x,t) = Y (t + x /v) k("x) 2 m ! One oscillator => m=M ! Sinusoidal waves: Moving to "right": y(x,t) = Asin (! t " # x + $0 ) Moving to "left": !x = !x L ! is mass per unit length ! y(x,t) = Asin (! t + " x + #0 ) ! Spring force T = k!x = tension T in the string
f = ! Wave speed: v =
April 21, 2006 ! = #f " (specific to medium of the wave)
9 ! v= k"x T = m /"x v 1 T = " " => can increase the pitch by tightening the string
Physics for Scientists&Engineers 1 Physics for Scientists&Engineers 1 April 21, 2006 ! 10 ! Elevator (1)
! An elevator repairman (mass 73 kg) sits on top of an elevator cabin of mass 655 kg inside a shaft of a skyscraper. The cabin is suspended by a 61 m long steel cable of mass 38 kg. He sends a signal to his colleague at the top of the elevator shaft by tapping the cable with his hammer. Elevator (2)
! Tension in cable from the weight of elevator + man
T = mg = (73 kg + 655 kg)(9.81 m/s 2 ) = 7142 N ! The linear mass density of the steel cable is = M 38 kg = = 0.623 kg/m L 61 m
v= T = 107 m/s ! Question: ! How long will it take for the wave pulse generated by the hammer tap to travel up the cable? ! The wave speed is ! The pulse travels up the 61 m long cable in
t= L 61 m = = 0.57 s v 107 m/s
Physics for Scientists&Engineers 1 12 April 21, 2006 Physics for Scientists&Engineers 1 11 April 21, 2006 ...
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This note was uploaded on 03/19/2008 for the course PHY 183 taught by Professor Wolf during the Spring '08 term at Michigan State University.
 Spring '08
 Wolf
 Physics

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