1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
|
{
"metadata": {
"name": "chapter24.ipynb"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 24: Mechanical Vibrations"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 24.24-1,Page No:596"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"# Initilization of variables\n",
"\n",
"f=0.1666666 # oscillations/second\n",
"x=8 # cm # distance from the mean position\n",
"pi=3.14\n",
"\n",
"# Calculations\n",
"\n",
"omega=2*pi*f\n",
"\n",
"# Amplitude is given by eq'n \n",
"r=sqrt((25*x**2)/16) # cm\n",
"\n",
"# Maximum acceleration is given as,\n",
"a_max=(pi/3)**2*10 # cm/s^2\n",
"\n",
"# Velocity when it is at a dist of 5 cm (assume s=5 cm) is given by\n",
"s=5 # cm\n",
"v=omega*(r**2-s**2)**0.5 # cm/s\n",
"\n",
"# Results\n",
"\n",
"print\"(a) The amplitude of oscillation is \",round(r,2),\"cm\"\n",
"print\"(b) The maximum acceleration is \",round(a_max,2),\"cm/s^2\"\n",
"print\"(c) The velocity of the particle at 5 cm from mean position is \",round(v,2),\"cm/s\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"(a) The amplitude of oscillation is 10.0 cm\n",
"(b) The maximum acceleration is 10.96 cm/s^2\n",
"(c) The velocity of the particle at 5 cm from mean position is 9.06 cm/s\n"
]
}
],
"prompt_number": 3
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 24.24-2,Page No:597"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"# Initilization of variables\n",
"\n",
"x_1=0.1 # m # assume the distance of the particle from mean position as (x_1 & x_2)\n",
"x_2=0.2# m \n",
"\n",
"# assume velocities as v_1 & v_2\n",
"\n",
"v_1=1.2 # m/s\n",
"v_2=0.8 # m/s\n",
"pi=3.14\n",
"\n",
"# Calculations\n",
"\n",
"# The amplitude of oscillations is given by dividing eq'n 1 by 2 as,\n",
"r=(0.064)**0.5 # m\n",
"omega=v_1*((r**2-x_1**2)**0.5) # radians/second\n",
"t=(2*pi)/omega # seconds\n",
"v_max=r*omega # m/s\n",
"\n",
"# let the max acceleration be a which is given as,\n",
"a=r*omega**2 # m/s^2\n",
"\n",
"# Results\n",
"\n",
"print\"(a) The amplitude of oscillations is \",round(r,3),\"m\"\n",
"print\"(b) The time period of oscillations is \",round(t,2),\"seconds\"\n",
"print\"(c) The maximum velocity is \",round(v_max,2),\"m/s\"\n",
"print\"(d) The maximum acceleration is \",round(a,2),\"m/s^2\"\n",
"# NOTE: the value of t is incorrect in the text book\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"(a) The amplitude of oscillations is 0.253 m\n",
"(b) The time period of oscillations is 1.22 seconds\n",
"(c) The maximum velocity is 1.31 m/s\n",
"(d) The maximum acceleration is 6.75 m/s^2\n"
]
}
],
"prompt_number": 14
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Exammple 24.24-5,Page No:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"# Initilization of variabes\n",
"\n",
"W=50 # N # weight\n",
"x_0=0.075 # m # amplitude\n",
"f=1 # oscillation/sec # frequency\n",
"pi=3.14\n",
"g=9.81 \n",
"\n",
"# Calculations\n",
"\n",
"omega=2*pi*f\n",
"K=(((2*pi)**2*W)/g)*(10**-2) # N/cm\n",
"\n",
"# let the total extension of the string be delta which is given as,\n",
"delta=(W/K)+(x_0*10**2) # cm\n",
"T=K*delta # N # Max Tension\n",
"v=omega*x_0 #m/s # max velocity\n",
"\n",
"# Results\n",
"\n",
"print\"(a) The stiffness of the spring is \",round(K,2),\"N/cm\"\n",
"print\"(b) The maximum Tension in the spring is \",round(T,2),\"N\"\n",
"print\"(c) The maximum velocity is \",round(v,2),\"m/s\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"(a) The stiffness of the spring is 2.01 N/cm\n",
"(b) The maximum Tension in the spring is 65.08 N\n",
"(c) The maximum velocity is 0.47 m/s\n"
]
}
],
"prompt_number": 15
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 24.24-10,Page No:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"# Initilization of variables\n",
"\n",
"l=1 # m # length of the simple pendulum\n",
"g=9.81 # m/s^2\n",
"pi=3.14\n",
"\n",
"# Calculations\n",
"\n",
"# Let t_s be the time period when the elevator is stationary\n",
"t_s=2*pi*(l/g)**0.5 #/ seconds\n",
"\n",
"# Let t_u be the time period when the elevator moves upwards. Then from eqn 1\n",
"t_u=2*pi*((l)/(g+(g/10)))**0.5 # seconds\n",
"\n",
"# Let t_d be the time period when the elevator moves downwards.\n",
"t_d=2*pi*(l/(g-(g/10)))**0.5 # seconds\n",
"\n",
"# Results\n",
"\n",
"print\"The time period of oscillation of the pendulum for upward acc of the elevator is \",round(t_u,2),\"seconds\"\n",
"print\"The time period of oscillation of the pendulum for downward acc of the elevator is \",round(t_d,2),\"seconds\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The time period of oscillation of the pendulum for upward acc of the elevator is 1.91 seconds\n",
"The time period of oscillation of the pendulum for downward acc of the elevator is 2.11 seconds\n"
]
}
],
"prompt_number": 17
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 24.24-11,Page No:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"# Initilization of variables\n",
"\n",
"t=1 # second # time period of the simple pendulum\n",
"g=9.81 # m/s^2\n",
"pi=3.14\n",
"\n",
"# Calculations\n",
"\n",
"# Length of pendulum is given as,\n",
"l=(t/(2*pi)**2)*g # m\n",
"\n",
"# Let t_u be the time period when the elevator moves upwards. Then the time period is given as,\n",
"t_u=2*pi*((l)/(g+(g/10)))**0.5 # seconds\n",
"\n",
"# Let t_d be the time period when the elevator moves downwards.\n",
"t_d=2*pi*(l/(g-(g/10)))**0.5 # seconds\n",
"\n",
"# Results\n",
"\n",
"print\"The time period of oscillation of the pendulum for upward acc of the elevator is \",round(t_u,2),\"seconds\"\n",
"print\"The time period of oscillation of the pendulum for downward acc of the elevator is \",round(t_d,2),\"seconds\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The time period of oscillation of the pendulum for upward acc of the elevator is 0.95 seconds\n",
"The time period of oscillation of the pendulum for downward acc of the elevator is 1.05 seconds\n"
]
}
],
"prompt_number": 19
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 24.24-12,Page No:"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"# Initilization of variables\n",
"\n",
"m=15 # kg # mass of the disc\n",
"D=0.3 # m # diameter of the disc\n",
"R=0.15 # m # radius\n",
"l=1 # m # length of the shaft\n",
"d=0.01 # m # diameter of the shaft\n",
"G=30*10**9 # N-m^2 # modulus of rigidity\n",
"pi=3.14\n",
"\n",
"# Calculations\n",
"\n",
"# M.I of the disc about the axis of rotation is given as,\n",
"I=(m*R**2)*0.5 # kg-m^2\n",
"\n",
"# Stiffness of the shaft\n",
"k_t=(pi*d**4*G)/(32*l) # N-m/radian\n",
"t=2*pi*(I/k_t)**0.5 # seconds\n",
"\n",
"# Results\n",
"\n",
"print\"The time period of oscillations of the disc is \",round(t,2),\"seconds\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The time period of oscillations of the disc is 0.48 seconds\n"
]
}
],
"prompt_number": 24
},
{
"cell_type": "code",
"collapsed": false,
"input": [],
"language": "python",
"metadata": {},
"outputs": [],
"prompt_number": 24
}
],
"metadata": {}
}
]
}
|
