import math
# Initilization of variables
f=1/6 # oscillations/second
x=8 # cm # distance from the mean position
# Calculations
omega=2*math.pi*f
# Amplitude is given by eq'n
r=math.sqrt((25*x**2)/16) # cm
# Maximum acceleration is given as,
a_max=(math.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*math.sqrt(r**2-s**2) # cm/s
# Results
print('(a) The amplitude of oscillation is %d cm'%r)
print('(b) The maximum acceleration is %f cm/s**2'%a_max)
print('(c) The velocity of the particle at 5 cm from mean position is %f cm/s'%v)
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
# Calculations
# The amplitude of oscillations is given by dividing eq'n 1 by 2 as,
r=math.sqrt(0.32/5) # m
omega=v_1/(math.sqrt(r**2-x_1**2)) # radians/second
t=(2*math.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 %f m'%r)
print('(b) The time period of oscillations is %f seconds'%t)
print('(c) The maximum velocity is %f m/s'%v_max)
print('(d) The maximum acceleration is %f m/s**2'%a) # the value of max acc is incorrect in the textbook
# NOTE: the value of t is incorrect in the text book
# The values may differ slightly due to decimal point accuracy
import math
# Initilization of variabes
W=50 # N # weight
x_0=0.075 # m # amplitude
f=1 # oscillation/sec # frequency
g=9.81
# Calculations
omega=2*math.pi*f
K=(((2*math.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 %f N/cm'%K)
print('(b) The maximum Tension in the spring is %f N'%T)
print('(c) The maximum velocity is %f m/s'%v)
import math
# Initilization of variables
l=1 # m # length of the simple pendulum
g=9.81 # m/s^2
# Calculations
# Let t_s be the time period when the elevator is stationary
t_s=2*math.pi*math.sqrt(l/g) #/ seconds
# Let t_u be the time period when the elevator moves upwards. Then from eqn 1
t_u=2*math.pi*math.sqrt((l)/(g+(g/10))) # seconds
# Let t_d be the time period when the elevator moves downwards.
t_d=2*math.pi*math.sqrt(l/(g-(g/10))) # seconds
# Results
print('The time period of oscillation of the pendulum for upward acc of the elevator is %f seconds'%t_u)
print('The time period of oscillation of the pendulum for downward acc of the elevator is %f seconds'%t_d)
import math
# Initilization of variables
t=1 # second # time period of the simple pendulum
g=9.81 # m/s^2
# Calculations
# Length of pendulum is given as,
l=(t/(2*math.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*math.pi*math.sqrt((l)/(g+(g/10))) # seconds
# Let t_d be the time period when the elevator moves downwards.
t_d=2*math.pi*math.sqrt(l/(g-(g/10))) # seconds
# Results
print('The time period of oscillation of the pendulum for upward acc of the elevator is %f seconds'%t_u)
print('The time period of oscillation of the pendulum for downward acc of the elevator is %f seconds'%t_d)
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
# Calculations
# M.I of the disc about the axis of rotation is given as,
I=(m*R**2)/2 # kg-m**2
# Stiffness of the shaft
k_t=(math.pi*d**4*G)/(32*l) # N-m/radian
t=2*math.pi*math.sqrt(I/k_t) # seconds
# Results
print('The time period of oscillations of the disc is %f seconds'%t)