## Description

Exercise 1: Chebyshev Polynomial of the First Kind

AIM:

Write a Python program that uses the NumPy Polynomial class to print a table of the first ten Chebyshev polynomials of the first kind. Here is the table generated by this program:

T_0(x) = 1

T_1(x) = x

T_2(x) = 2*x^2 – 1

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Exercise 2: Sound Intensity from a Point Source

In an experiment, Mary measured the variation of the intensity I of the sound produced by a point source with the distance r from the source. Here is her measurement result:

r (m) 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

I (10โ5 W/m2) 0.987 0.662 0.525 0.373 0.308 0.262 0.191 0.184

On the other hand, physical theories tell us that for a point source of sound of power P, the sound intensity I and the distance r from the source are related by:

P

I = 4 ๏ฐr2

Write a Python program that uses the np.linalg method lstsq to find the best least-square fit of

ln I = m ln r + k

for the given data and then display the fitting result together with the theoretical prediction. Your program should output a table of the values of the fitting parameters m and k found from the fitting and their theoretical values as well as the root-mean-square of the residual of the fitting. Assume that the point source emits sound with a power of P = 4๏ฐร10โ5 W.

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Exercise 3: Forced Vibration with Damping

A small block of mass m suspended vertically by a spring with spring constant k is driven by an external force F(t) = F0 cos(๏ทt). The block is moving in a viscous medium with a damping force of the form โbv where b > 0 is the damping constant and v is its instantaneous velocity. Taking downward as the positive direction, the vibration of the block is modeled by the differential equation:

d2x dx

m 2 + b dt + kx = F0 cos(๏ทt) dt

where x(t) is the displacement of the block from its equilibrium position at time t. It can be shown that the steady state solution (i. e. x(t) when time t โ ๏ฅ) is

xs(t) = (MF0/k) cos(๏ทt โ๏ฆ)

In this formula, M is the magnification ratio and ๏ฆ is the phase lag defined by

1 2๏บ(๏ท/๏ท

M = 2/๏ท02)2 + 4๏บ (๏ท/๏ท0) , ๏ฆ= tanโ1 [1 โ (๏ท/๏ท00))2],

โ(1 โ ๏ท 2 2

where ๏ท0 = โk/m is the natural frequency and ๏บ is the damping ratio. Write a Python program that uses the Matplotlib Axes class method plot to plot the magnification ratio M over the interval of frequency ratio ๏ท/๏ท0 from 0 to 2.0 for damping ratio ๏บ = 0.1, 0.2, 0.4, 0.6, and 0.8, respectively, on the same graph. You should label your graph with proper axis labels, title, and legends. From your graph, you can observe how the peak value of M depends on ๏บ, i. e. the effect of damping on the resonance frequency of the block.

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Exercise 4: Employees in Hong Kong’s Construction Industry

Year Number of Employees in Thousands Share of the Employees in the Labour Force

2011 277.0 7.75 %

2012 290.1 7.93 %

2013 309.0 8.30 %

2014 309.7 8.27 %

2015 316.7 8.39 %

2017 342.0 8.95 %

Write a Python program that uses Matplotlib Axes class method twinx to produce a bar chart of the number of employees in Hong Kongโs construction industry and a line plot of the percentage share of these employees in the labour force as a function of year on the same graph. You should label your graph with proper axis labels, title, and legends.

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