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+{
+ "metadata": {
+ "name": "",
+ "signature": ""
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+ {
+ "cells": [
+ {
+ "cell_type": "heading",
+ "level": 1,
+ "metadata": {},
+ "source": [
+ "Chapter 8, Vibrations of strings & membranes"
+ ]
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 1, page 317"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "from __future__ import division\n",
+ "from numpy import sqrt\n",
+ "# Speed\n",
+ "#given data :\n",
+ "m1=0.1 # in kg\n",
+ "g=9.81 # in m/s**2\n",
+ "T=m1*g # N\n",
+ "A=10**-6 # in m**2\n",
+ "p=9.81*10**3 # in kg/m**3\n",
+ "m=A*p # in kg/m\n",
+ "v=sqrt(T/m) \n",
+ "print \"The speed of transverse waves, v = %0.f m/s \" %v"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The speed of transverse waves, v = 10 m/s \n"
+ ]
+ }
+ ],
+ "prompt_number": 2
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 2, page 318"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# tensile stress\n",
+ "#given data :\n",
+ "p=8000 # in kg/m**3\n",
+ "v=340 # in m/s\n",
+ "TbyA=v**2*p*10**-2 \n",
+ "print \"Tensile stress = %0.2e N/m^2 \" %TbyA"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Tensile stress = 9.25e+06 N/m^2 \n"
+ ]
+ }
+ ],
+ "prompt_number": 3
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 3, page 323"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# Tension\n",
+ "#given data :\n",
+ "M=2*10**-3 # in kg\n",
+ "l=35*10**-2 # in m\n",
+ "n=500 # in Hz\n",
+ "m=M/l # in kg/m\n",
+ "T=4*n**2*l**2*m \n",
+ "print \"Tension, T = %0.f N \" %T"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Tension, T = 700 N \n"
+ ]
+ }
+ ],
+ "prompt_number": 5
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 4, page 324"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# Frequency\n",
+ "#given data :\n",
+ "T=625 # in N\n",
+ "T1=100 # in N\n",
+ "l=1/2 \n",
+ "n=240 # in Hz\n",
+ "n1=1/l*(sqrt(T1/T))*n \n",
+ "print \"The frequency, n1 = %0.f Hz \" %n1"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The frequency, n1 = 192 Hz \n"
+ ]
+ }
+ ],
+ "prompt_number": 6
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 5, page 324"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# initial tension\n",
+ "rt=2/3 #ratio\n",
+ "mi=5 #kg wt\n",
+ "M=((1/rt)**2)-1 #\n",
+ "mo=mi/M #kg wt\n",
+ "print \"Initial tension in string = %0.2f kg-wt \" % mo"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Initial tension in string = 4.00 kg-wt \n"
+ ]
+ }
+ ],
+ "prompt_number": 7
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 6, page 325"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# speed,stress and change in frequency\n",
+ "n=175 #Hz\n",
+ "l=1.5 #m\n",
+ "v=2*n*l #m/s\n",
+ "d=7.8*10**3 #kg/m**3\n",
+ "st=v**2*d #N/m**2\n",
+ "per=3 #% increament\n",
+ "T=1 #assume\n",
+ "td=(1+per/100)*T #\n",
+ "x=(((1/2)*(per/100))) #\n",
+ "td=x*100 #\n",
+ "print \"Velocity = %0.2f m/s \" % v\n",
+ "print \"Stress = %0.2e N/m^2 \" %st\n",
+ "print \"Percentage change in frequency = %0.1f %% \" %td"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Velocity = 525.00 m/s \n",
+ "Stress = 2.15e+09 N/m^2 \n",
+ "Percentage change in frequency = 1.5 % \n"
+ ]
+ }
+ ],
+ "prompt_number": 10
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 7, page 326"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# Frequency\n",
+ "#given data :\n",
+ "l=.50 # in m\n",
+ "m1=25 # in kg\n",
+ "m2=1.44*10**-3 # in kg\n",
+ "g=9.81 # in m/s**2\n",
+ "T=m1*g \n",
+ "m=m2/l \n",
+ "p=2 \n",
+ "n=(p/(2*l))*sqrt(T/m) \n",
+ "print \"The frequency, n = %0.1f \" %n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "The frequency, n = 583.6 \n"
+ ]
+ }
+ ],
+ "prompt_number": 12
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 8, page 326"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# frequency\n",
+ "l1=90 #cm\n",
+ "d1=0.05 #cm\n",
+ "d2=0.0625 #cm\n",
+ "l2=60 #cm\n",
+ "n1=200 #Hz\n",
+ "n2=((l1*d1*n1)/(l2*d2)) #Hz\n",
+ "print \"Frequency = %0.2f Hz \" % n2"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Frequency = 240.00 Hz \n"
+ ]
+ }
+ ],
+ "prompt_number": 13
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 9, page 327"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# tension\n",
+ "n21=3/2 #\n",
+ "r21=3/4 #\n",
+ "t1=2.048 #kg. wt\n",
+ "t2=(n21*r21)**2*t1 #kg weight\n",
+ "n31=9/4 #\n",
+ "r31=2/4 #\n",
+ "t3=(n31*r31)**2*t1 #kg-weight\n",
+ "n41=27/8 #\n",
+ "r41=1/4 #\n",
+ "t4=(n41*r41)**2*t1 #kg-weight\n",
+ "print \"Tension, T2 = %0.3f kg weight\"%t2\n",
+ "print \"Tension, T3 = %0.3f kg weight\"%t3\n",
+ "print \"Tension, T4 = %0.3f kg weight\"%t4"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Tension, T2 = 2.592 kg weight\n",
+ "Tension, T3 = 2.592 kg weight\n",
+ "Tension, T4 = 1.458 kg weight\n"
+ ]
+ }
+ ],
+ "prompt_number": 16
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 10, page 328"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "from math import sqrt\n",
+ "# velocity\n",
+ "l1=20 #cm\n",
+ "v1=600 #cm**-1\n",
+ "n1=v1/4 #\n",
+ "v1=2*n1*l1*10**-2 #m/sec\n",
+ "v2=sqrt(2)*v1 #m/s\n",
+ "print \"Velocity of the waves = %0.f m/s \" %v1\n",
+ "print \"Velocity of waves when tension of the string is doubled = %.f m/s \" %round(v2)"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Velocity of the waves = 60 m/s \n",
+ "Velocity of waves when tension of the string is doubled = 85 m/s \n"
+ ]
+ }
+ ],
+ "prompt_number": 17
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 11, page 331"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# frequency\n",
+ "nb=6 #beats\n",
+ "l1=20 #cm\n",
+ "l2=21 #cm\n",
+ "x=l2/l1 #\n",
+ "n=(x*nb+nb)/(x-1) #\n",
+ "print \"Frequency = %0.f Hz \" %n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Frequency = 246 Hz \n"
+ ]
+ }
+ ],
+ "prompt_number": 18
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 12, page 331"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# frequency\n",
+ "nb=4 #beats\n",
+ "l1=70 #cm\n",
+ "l2=70-1 #cm\n",
+ "x=l2/l1 #\n",
+ "n=(x*nb)/(1-x) #\n",
+ "print \"Frequency = %0.f Hz \" %n"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Frequency = 276 Hz \n"
+ ]
+ }
+ ],
+ "prompt_number": 19
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 13, page 332"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "# length\n",
+ "n123=1/3/15 #\n",
+ "tl=105 #cm\n",
+ "l123=15/5/1 #\n",
+ "k=tl/21 #\n",
+ "l1=15*k #cm\n",
+ "l2=5*k #cm\n",
+ "l3=k #cm\n",
+ "print \"l1 length = %0.f cm\"%l1\n",
+ "print \"l2 length = %0.f cm\"%l2\n",
+ "print \"l3 length = %0.f cm\"%l3\n",
+ "#length l2 is calculated wrong in the textbook"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "l1 length = 75 cm\n",
+ "l2 length = 25 cm\n",
+ "l3 length = 5 cm\n"
+ ]
+ }
+ ],
+ "prompt_number": 20
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 15, page 355"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "from math import sqrt\n",
+ "# frequency\n",
+ "l=2.5 #m\n",
+ "m1=0.001 #kg\n",
+ "tn=4 #N\n",
+ "m=m1/l #kg/m\n",
+ "n=((1/(2*l))*sqrt(tn/m)) #Hz\n",
+ "print \"Frequency = %0.2f Hz \" %n\n",
+ "print \"Frequencies stopped are\",5*n,\"Hz, \",10*n,\"Hz, \",15*n,\"Hz\""
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Frequency = 20.00 Hz \n",
+ "Frequencies stopped are 100.0 Hz, 200.0 Hz, 300.0 Hz\n"
+ ]
+ }
+ ],
+ "prompt_number": 22
+ },
+ {
+ "cell_type": "heading",
+ "level": 2,
+ "metadata": {},
+ "source": [
+ "Example 16, page 356"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "collapsed": false,
+ "input": [
+ "from math import pi\n",
+ "# frequency\n",
+ "l=1 #m\n",
+ "m1=0.5 #kg\n",
+ "tn=200 #N\n",
+ "m=m1/l #kg/m\n",
+ "n=((1/(2*l))*sqrt(tn/m)) #Hz\n",
+ "print \"Frequency = %0.2f Hz \" %n\n",
+ "w=2*pi*n #\n",
+ "print \"Ratio of three frequencies is %0.1f:%0.1f:%0.1f\"%(w,2*w,3*w)"
+ ],
+ "language": "python",
+ "metadata": {},
+ "outputs": [
+ {
+ "output_type": "stream",
+ "stream": "stdout",
+ "text": [
+ "Frequency = 10.00 Hz \n",
+ "Ratio of three frequencies is 62.8:125.7:188.5\n"
+ ]
+ }
+ ],
+ "prompt_number": 25
+ }
+ ],
+ "metadata": {}
+ }
+ ]
+}