{
"metadata": {
"name": "",
"signature": "sha256:c2cfd770b64ff6a5a34b3865d707320d0a14506274eb94013c7282c3c39c74ee"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"3 : Mechanics of rigid body"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 1, Page number A 3.32"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"t = 20 # tow is given in N-m\n",
"k = 0.5 # radius of gyration is given in meters\n",
"m = 10 # mass is given in Kgs\n",
"# alpha = ? alpha is angular acceleration\n",
"#alpha = tow / (((gyration)**2) * mass)\n",
"\n",
"#calculation\n",
"alpha = t / ((k**2) * m)\n",
"\n",
"#result\n",
"print \"the angular acceleration is\", alpha ,\"rad /(sec**2)\"\n",
"print \"answer given in the book is wrong\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the angular acceleration is 8.0 rad /(sec**2)\n",
"answer given in the book is wrong\n"
]
}
],
"prompt_number": 2
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 2, Page number A 3.32"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"pi = 22/7\n",
"m = 6 * 10**24 # mass is given in kgs\n",
"r = 6.4 * 10**6 # radius is given in meters \n",
"r1 = 1.5 * 10**11 # orbital radius in meters\n",
"t = 24*60*60 # number of seconds in a day\n",
"T = 365*24*60*60 # number of seconds in a year\n",
"\n",
"I = (2/5) * m * r**2 # inertia in kg-m**2\n",
"I1 = m * r1**2 # inertia in kg-m**2\n",
"omega = (2* pi) / t # angular velocity in rad / sec\n",
"omega1 = (2* pi) / T\n",
"\n",
"#calculation\n",
"L = I * omega # spin angular momentum (Kg -m**2) / sec\n",
"L1 = I1 * omega1 # Orbital angular momentum ((Kg -m**2) / sec)\n",
"\n",
"#Result\n",
"print \"the spin angular momentum is \", round(L/10**33,3), \"*10**33 Kg m**2 / sec\"\n",
"print \"answer varies due to rounding off errors\"\n",
"print \"the spin angular momentum is \", round(L1/10**40,2), \"*10**40 Kg m**2 / sec\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the spin angular momentum is 7.152 *10**33 Kg m**2 / sec\n",
"answer varies due to rounding off errors\n",
"the spin angular momentum is 2.69 *10**40 Kg m**2 / sec\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 4, Page number A 3.33"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"m = 0.1 # mass is converted into kgs from grams\n",
"d = 0.01 # diameter is given in meters\n",
"r = d/2 # radius is half of diameter\n",
"v = 0.05 # velocity is given in m / s\n",
"\n",
"I = (2/5) * m *r**2 # inertia is given in Kg - m**2\n",
"omega = v/r # angular velocity is given in rad/sec\n",
"\n",
"#calculation\n",
"KE = (1/2)* I * omega**2 # kinetic energy is given in joules\n",
"\n",
"#Result\n",
"print \"the kinetic energy of the sphere is\", round(KE*10**5), \"*10**-5 j\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the kinetic energy of the sphere is 5.0 *10**-5 j\n"
]
}
],
"prompt_number": 7
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 5, Page number A 3.34"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"d = .02 # diameter is converted to meters from centimeters\n",
"r = d/2 # radius is half of diameter\n",
"I = 2 * 10**(-6) # inertia is given in kg - mtr**2\n",
"v = .05 # velocity is converted into mtrs/sec from cms / sec\n",
"omega = v/r # omega is angular velocity in rad / sec\n",
"\n",
"#calculation\n",
"KE = (1/2) * I * omega**2 # kinetic energy is calculated in joules\n",
"\n",
"#Result \n",
"print \"the kinetic energy calculated is\" ,KE*10**6, \"*10**-6 j\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the kinetic energy calculated is 25.0 *10**-6 j\n"
]
}
],
"prompt_number": 12
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 6, Page number A 3.34"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"m = 100 # mass is given in Kgs\n",
"r = 1 # radius is given in meters\n",
"n = 120 # number of rotations is equall to 120 rotations per minute\n",
"pi = 3.14\n",
"t = 60 # time t is given in seconds as the rotations are given according to minute\n",
"\n",
"#calculation\n",
"I = (1/2) * m * r**2 # inertia is calculated in Kg - mtr**2\n",
"omega = (2*pi*n)/t # omega is angular velocity\n",
"\n",
"KE = (1/2) * I * omega**2 # kinetic energy is calculated in joules\n",
"\n",
"#Result\n",
"print \"the kinetic energy is \", round(KE), \"j\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the kinetic energy is 3944.0 j\n"
]
}
],
"prompt_number": 13
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 8, Page number A 3.35"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"f = 40 # frequency is given in revolutions per sec\n",
"b = 0.1 # base is given in cetimeters which is converted into meters\n",
"# inertia (I) is given as (3/10)* m * r**2\n",
"\n",
"# omega = (m*g*r)/L\n",
"\n",
"#Calculation\n",
"omega = (10*9.8*20*10**(-2))/(4*25*10**(-4) * 6.28 * 40) # angular velocity is cal in rad /sec\n",
"\n",
"#reuslt\n",
"print \"the angular velocity\", round(omega,4), \"rad/sec\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the angular velocity 7.8025 rad/sec\n"
]
}
],
"prompt_number": 14
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 9, Page number A 3.36"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"m = 1.5 # mass is given in Kgs\n",
"k = 0.3 # radius of gyration is given in mtrs\n",
"n = 240 # number of revolutions per miute\n",
"t = 60 # time is taken in seconds as the revolutioni is given per minute\n",
"L = 0.1 # pivoted point length is given in meters\n",
"g = 9.8 # gravitational constant is 9.8 mtrs/sec**2\n",
"\n",
"#calculation\n",
"omega = (2*pi*n) / t # omega is calculated in rad /sec\n",
"\n",
"omegap = (g * L) / (k**2 *omega)\n",
"\n",
"#result\n",
"print \"the precessional speed of the wheel is\", round(omegap,2), \"red / sec\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the precessional speed of the wheel is 0.43 red / sec\n"
]
}
],
"prompt_number": 15
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 10, Page number A 3.36"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"m = 1 # mass is given in kgs\n",
"r = 0.1 # radius is given in meters\n",
"omega = 20 * 3.14 # omega(angular velocity) is given in rad/ sec\n",
"I = (1/2)*m*r**2 # inertia is calcuated in kg - m**2\n",
"\n",
"#calculation\n",
"\n",
"L= I* omega #L agular momentum is calculated in m**2 / sec\n",
"KE = (1/2) * I * (omega**2) # kineticc energy is calculated in joules\n",
"\n",
"#result\n",
"print \"the angular momentum is\", L, \"m**2 / sec\"\n",
"print \"the kinetic energy is\", KE, \"j\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the angular momentum is 0.314 m**2 / sec\n",
"the kinetic energy is 9.8596 j\n"
]
}
],
"prompt_number": 18
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 12, Page number A 3.38"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"m = 0.2 # mass is given in kgs\n",
"r = 0.5 # radius is given in meters\n",
"a = 0.2 # rate of change of velocity is acceleration which ig given in mtr / sec**2\n",
"\n",
"# calculation\n",
"\n",
"# tow is rate of change of angular momentum\n",
"# L = m* v * r\n",
"# by differentiating L we have to differentiate v that is velocity which gives us acceleration a \n",
"\n",
"t = m * r * a # tow is calculated in N/m\n",
"\n",
"#Result\n",
"print \"the torque acting is\",t , \"N - m\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the torque acting is 0.02 N - m\n"
]
}
],
"prompt_number": 19
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 13, Page number A 3.38"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"m = 2 # mass is given in kgs\n",
"r = 0.5 # radius is given in meters\n",
"v = 24*1000/(60*60) # velocity is given in Km / hr which is cinverted into mtrs / sec\n",
"t = 0.1 # time is given in sec\n",
"omega = v/r # angular velocity is calc in rad/sec\n",
"theta = 1 # angular change\n",
"\n",
"# calculation\n",
"L = m * r**2 * omega # angular momentum Kg -m**2/sec\n",
"#t = d/dt(L)\n",
"t= L * theta/t # torque in N -m\n",
"\n",
"#result\n",
"print \"the angular momentum\",round(L,3),\"in KG - m**2/sec\"\n",
"print \"the torque required is\",round(t,2),\"N - m\" "
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the angular momentum 6.667 in KG - m**2/sec\n",
"the torque required is 66.67 N - m\n"
]
}
],
"prompt_number": 22
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 14, Page number A 3.38"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"KE = 80 # kinetic energy is calculated in joules\n",
"I = 8 * 10 **(-7) # inertia is given in kg-m**2\n",
"\n",
"# calculation\n",
"L = (2* I * KE)**(0.5) # the angular momentum is given in Kg -m**2/sec\n",
"\n",
"# result\n",
"print \"the angular momentum is\", round(L*10**2,2) , \"*10**-2 Kg - m**2/sec\""
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the angular momentum is 1.13 *10**-2 Kg - m**2/sec\n"
]
}
],
"prompt_number": 27
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example number 15, Page number A 3.39\n"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#import modules\n",
"import math\n",
"from __future__ import division\n",
"\n",
"#given data\n",
"n = 30 # numb of revolution is given in rev / sec\n",
"t = 1 # time t is in seconds\n",
"theta = 30 # angle theta is 30 degrees\n",
"m = 0.5 # mass is given in Kgs \n",
"I = 5 * 10**-4 # rotational inertia is given in Kg -m**2\n",
"l = 0.04 # length pivoted from the center of mass is given in cms which is converted into meters\n",
"pi= 3.14\n",
"g = 9.8 # gravitational const is given in m/sec**2\n",
"\n",
"#calculation\n",
"omega = (2*pi*n)/t # angular velocity is calculated in rad/ sec\n",
"omegap = (m*g*l)/(I* omega) # angular velocity of precession is calculated in rad/ sec\n",
"\n",
"#Result\n",
"print \"the angular velocity of precession is\",round(omegap,2),\"rad/sec\" "
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"the angular velocity of precession is 2.08 rad/sec\n"
]
}
],
"prompt_number": 24
}
],
"metadata": {}
}
]
}