Chapter 24: Mechanical Vibrations¶

import math

# Initilization of variables

f=0.1666666 # oscillations/second
x=8 # cm # distance from the mean position
pi=3.14

# Calculations

omega=2*pi*f

# Amplitude is given by eq'n 
r=sqrt((25*x**2)/16) # cm

# Maximum acceleration is given as,
a_max=(pi/3)**2*10 # cm/s^2

# Velocity when it is at a dist of 5 cm (assume s=5 cm) is given by
s=5 # cm
v=omega*(r**2-s**2)**0.5 # cm/s

# Results

print"(a) The amplitude of oscillation is ",round(r,2),"cm"
print"(b) The maximum acceleration is ",round(a_max,2),"cm/s^2"
print"(c) The velocity of the particle at 5 cm from mean position  is ",round(v,2),"cm/s"

import math

# Initilization of variables

x_1=0.1 # m # assume the distance of the particle from mean position as (x_1 & x_2)
x_2=0.2# m 

# assume velocities as v_1 & v_2

v_1=1.2 # m/s
v_2=0.8 # m/s
pi=3.14

# Calculations

# The amplitude of oscillations is given by dividing eq'n 1 by 2 as,
r=(0.064)**0.5 # m
omega=v_1*((r**2-x_1**2)**0.5) # radians/second
t=(2*pi)/omega # seconds
v_max=r*omega # m/s

# let the max acceleration be a which is given as,
a=r*omega**2 # m/s^2

# Results

print"(a) The amplitude of oscillations is ",round(r,3),"m"
print"(b) The time period of oscillations is ",round(t,2),"seconds"
print"(c) The maximum velocity is ",round(v_max,2),"m/s"
print"(d) The maximum acceleration is ",round(a,2),"m/s^2"
# NOTE: the value of t is incorrect in the text book

import math

# Initilization of variabes

W=50 # N # weight
x_0=0.075 # m # amplitude
f=1 # oscillation/sec # frequency
pi=3.14
g=9.81 

# Calculations

omega=2*pi*f
K=(((2*pi)**2*W)/g)*(10**-2) # N/cm

# let the total extension of the string be delta which is given as,
delta=(W/K)+(x_0*10**2) # cm
T=K*delta # N # Max Tension
v=omega*x_0 #m/s # max velocity

# Results

print"(a) The stiffness of the spring is ",round(K,2),"N/cm"
print"(b) The maximum Tension in the spring is ",round(T,2),"N"
print"(c) The maximum velocity is ",round(v,2),"m/s"

import math

# Initilization of variables

l=1 # m # length of the simple pendulum
g=9.81 # m/s^2
pi=3.14

# Calculations

# Let t_s be the time period when the elevator is stationary
t_s=2*pi*(l/g)**0.5 #/ seconds

# Let t_u be the time period when the elevator moves upwards. Then from eqn 1
t_u=2*pi*((l)/(g+(g/10)))**0.5 # seconds

# Let t_d be the time period when the elevator moves downwards.
t_d=2*pi*(l/(g-(g/10)))**0.5 # seconds

# Results

print"The time period of oscillation of the pendulum for upward acc of the elevator is ",round(t_u,2),"seconds"
print"The time period of oscillation of the pendulum for downward acc of the elevator is ",round(t_d,2),"seconds"

import math

# Initilization of variables

t=1 # second # time period of the simple pendulum
g=9.81 # m/s^2
pi=3.14

# Calculations

# Length of pendulum is given as,
l=(t/(2*pi)**2)*g # m

# Let t_u be the time period when the elevator moves upwards. Then the time period is given as,
t_u=2*pi*((l)/(g+(g/10)))**0.5 # seconds

# Let t_d be the time period when the elevator moves downwards.
t_d=2*pi*(l/(g-(g/10)))**0.5 # seconds

# Results

print"The time period of oscillation of the pendulum for upward acc of the elevator is ",round(t_u,2),"seconds"
print"The time period of oscillation of the pendulum for downward acc of the elevator is ",round(t_d,2),"seconds"

import math

# Initilization of variables

m=15 # kg # mass of the disc
D=0.3 # m # diameter of the disc
R=0.15 # m # radius
l=1 # m # length of the shaft
d=0.01 # m # diameter of the shaft
G=30*10**9 # N-m^2 # modulus of rigidity
pi=3.14

# Calculations

# M.I of the disc about the axis of rotation is given as,
I=(m*R**2)*0.5 # kg-m^2

# Stiffness of the shaft
k_t=(pi*d**4*G)/(32*l) # N-m/radian
t=2*pi*(I/k_t)**0.5 # seconds

# Results

print"The time period of oscillations of the disc is ",round(t,2),"seconds"