{
"metadata": {
"name": ""
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 7: Waves and Oscillations"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.1, Page 7.22"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import pi, sin, sqrt\n",
"\n",
"# Given \n",
"E = 1.024e-3 # total energy of particle in J\n",
"T = 2 * pi # time period of S.H.M. in sec\n",
"x = 0.08 * sqrt(2) # distance of partile in meter\n",
"t = pi / 4 # time in second\n",
"\n",
"#Calculations\n",
"A = x / sin((2 * pi * t) / T)\n",
"M = (E * T**2) / (2 * pi**2 * A**2)\n",
"\n",
"#Result\n",
"print \"Amplitude = %.2f meter\\nMass of particle = %.f g\"%(A,M/1e-3)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Amplitude = 0.16 meter\n",
"Mass of particle = 80 g\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.2, Page 7.22"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import pi\n",
"\n",
"# Given \n",
"A = 0.05 # amplitude in meter\n",
"T = 10 # time period of S.H.M. in sec\n",
"\n",
"#Calculations\n",
"v = (A * 2 * pi) / T\n",
"\n",
"#Result\n",
"print \"Maximum amplitude of velocity = %.4f meter/sec\"%v"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Maximum amplitude of velocity = 0.0314 meter/sec\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.3, Page 7.23"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import pi, sqrt\n",
"\n",
"# Given \n",
"E = 9 # total energy of particle in J\n",
"U = 5 # potential energy in J\n",
"A = 1 # amplitude in meter\n",
"m = 2. # mass of harmonic oscillator in kg\n",
"\n",
"#Calculations\n",
"kE = E - U# calculation for kinetic energy\n",
"k = (2 * kE) / A**2# calculation for force constant\n",
"T = (2 * pi) * sqrt(m / k)# calculation for time period\n",
"\n",
"#Result\n",
"print \"Force constant = %.f J/m\\nTime period = %.2f sec\"%(k,T)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Force constant = 8 J/m\n",
"Time period = 3.14 sec\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.4, Page 7.23"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import asin, sin, pi\n",
"\n",
"# Given \n",
"A = 0.06 # amplitude in meter\n",
"T = 6 # time period of S.H.M. in sec\n",
"x = 0.03 # position of particle in meter\n",
"\n",
"#Calculations\n",
"delta = asin(1) # by the formula x=Asin(wt+delta) and (at t = 0,x=A) \n",
"t = x / (A * sin(((2 * pi) / T) + delta))\n",
"\n",
"#Result\n",
"print \"Time taken by the particle = %.f sec\"%t"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Time taken by the particle = 1 sec\n"
]
}
],
"prompt_number": 7
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.5, Page 7.24"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import pi\n",
"\n",
"# Given \n",
"A = 0.05# amplitude in meter\n",
"T = 10 * pi # time period of s.h.m. in sec\n",
"\n",
"#Calculations\n",
"v = A * (2 * pi / T)\n",
"a = A * (2 * pi / T)**2\n",
"\n",
"#Result\n",
"print \"Maximum velocity = %.e meter/sec\\nacceleration = %.e m/sec^2\"%(v,a)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Maximum velocity = 1e-02 meter/sec\n",
"acceleration = 2e-03 m/sec^2\n"
]
}
],
"prompt_number": 9
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.6, Page 7.24"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import pi\n",
"\n",
"# Given \n",
"A = 0.06# amplitude in meter\n",
"T = 10 * pi # time period of s.h.m. in sec\n",
"\n",
"#Calculation\n",
"v = A * (2 * pi / T)\n",
"\n",
"#Result\n",
"print \"Maximum velocity = %.1e meter/sec\"%v"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Maximum velocity = 1.2e-02 meter/sec\n"
]
}
],
"prompt_number": 10
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.7, Page 7.24"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import sqrt, pi\n",
"\n",
"# Given \n",
"k = 16 # stiffness constant of spring n/m\n",
"m = 1 # mass of particle in kg\n",
"\n",
"#Calculations\n",
"n = sqrt(k / m) / (2 * pi)\n",
"\n",
"#Result\n",
"print \"natural frequency = %.2f Hz\"%n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"natural frequency = 0.64 Hz\n"
]
}
],
"prompt_number": 11
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.8, Page 7.25"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import sqrt, pi\n",
"\n",
"# Given \n",
"l = 1 # length of pendulum in meter\n",
"m = 2 # mass of particle in kg\n",
"g = 9.8 # acceleration due to gravity in m/sec^2\n",
"\n",
"#Calculation\n",
"T = 2 * pi * sqrt(l / g)\n",
"\n",
"#Result\n",
"print \"The time period of pendulum = %.f sec\"%T"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The time period of pendulum = 2 sec\n"
]
}
],
"prompt_number": 12
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.9, Page 7.25"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import sqrt, pi\n",
"\n",
"# Given \n",
"m = 100. # mass of particle in gm\n",
"\n",
"#Calculation\n",
"n = (1 / (2 * pi)) * sqrt(10 / m) # by using given formula \n",
"\n",
"#Result\n",
"print \"Frequency = %.2f Hz\"%n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Frequency = 0.05 Hz\n"
]
}
],
"prompt_number": 13
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.10, Page 7.25"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import sqrt, pi\n",
"\n",
"# Given \n",
"f = 3 # acceleration of pendulum in m/sec^2\n",
"l = 1 # length of pendulum in meter\n",
"g = 9.8 # acceleration due to gravity in m/sec^2\n",
"\n",
"#calculatiom\n",
"T = 2 * pi * sqrt(l / (g + f))\n",
"\n",
"#Result\n",
"print \"Time period of pendulum = %.2f sec\"%T"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Time period of pendulum = 1.76 sec\n"
]
}
],
"prompt_number": 14
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.11, Page 7.26"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import sqrt, pi\n",
"\n",
"# Given \n",
"x = 0.3 # stretch in spring in meter\n",
"m1 = 6 # mass of first body in kg\n",
"m2 = 1 # mass of second body in kg\n",
"g = 9.8 # gravitational acceleration of earth in m/sec^2\n",
"\n",
"#Calculations\n",
"k = (m1 * g) / x\n",
"T = (2 * pi) * sqrt(m2 / k)\n",
"\n",
"#Result\n",
"print \"Time period of motion = %.2f sec \"%T"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Time period of motion = 0.45 sec \n"
]
}
],
"prompt_number": 15
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.12, Page 7.26"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import sqrt, pi\n",
"\n",
"# Given \n",
"x = 0.1 # compression in spring in m\n",
"F = 10 # restoring force in N\n",
"m = 4 # mass of body in kg\n",
"g = 9.8 # acceleration due to gravity in m/sec^2\n",
"\n",
"#Calculations\n",
"k = F / x\n",
"F_ = m * g\n",
"x_ = F_ / k\n",
"T = (2 * pi) * sqrt(m / k)\n",
"\n",
"#Result\n",
"print \"Time period of motion = %.2f sec \\nCompression of the spring due to the weight of the body = %.3f m \"%(T,x_)\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Time period of motion = 1.26 sec \n",
"Compression of the spring due to the weight of the body = 0.392 m \n"
]
}
],
"prompt_number": 16
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.13, Page 7.26"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import exp\n",
"\n",
"# Given \n",
"t = 50. # relaxation time in sec\n",
"r = 1 / exp(1) # falls in amplitude and energy\n",
"\n",
"#Calculations\n",
"s = 1 / (2 * t)\n",
"T = 1 / s # by using formula A=A_exp(-st) and using r=A/A_\n",
"\n",
"#Result\n",
"print \"Time = %.f sec\"%T\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Time = 100 sec\n"
]
}
],
"prompt_number": 22
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.14, Page 7.27"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"from math import pi, exp\n",
"\n",
"# Given \n",
"n = 260 # frequency in Hz\n",
"Q = 2000 # quality factor\n",
"r = 1 / (exp(1)**2) # decrease in amplitude \n",
"\n",
"#Calculations\n",
"tou = Q / (2 * pi * n)\n",
"t = 2 * tou # by using formula A=A_exp(-st) and using r=A/A_\n",
"\n",
"#Result\n",
"print \"Time = %.3f sec\"%t\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"Time = 2.449 sec\n"
]
}
],
"prompt_number": 17
}
],
"metadata": {}
}
]
}