# How many poles does the Laplace Transform of a square wave have? a) 0 b) 1 c) 2 d) Infinitely Manhy

Question
Laplace transform
How many poles does the Laplace Transform of a square wave have?
a) 0
b) 1
c) 2
d) Infinitely Manhy

2021-02-27
Solution:
The Fourier series for square wave is $$f(x)=\frac{4}{\pi} \sum_{i=1,3,5,\dots}^\infty \frac{1}{n}\sin\left(\frac{n\pi x}{L}\right)$$
The Laplace transform is $$L\left\{\sin(ax)\right\}=\frac{a}{x^2+a^2}$$
The function is $$f(x)=\frac{4}{\pi}\left[\sin\left(\frac{\pi x}{L}\right)+\frac{1}{3}\sin\left(\frac{3\pi x}{L}\right)+\frac{1}{5}\sin\left(\frac{5\pi x}{L}\right)+\dots\right]$$
Apply Laplace transform:
Conclusion:
$$L{f(x)}=\frac{4}{\pi}\left[L\left\{\sin\left(\frac{\pi x}{L}\right)\right\}+\frac{1}{3}L\left\{\sin\left(\frac{3\pi x}{L}\right)\right\}+\frac{1}{5}L\left\{\sin\left(\frac{5\pi x}{L}\right)\right\}+\dots\right]$$
$$=\frac{4}{\pi}\left[\frac{\frac{\pi}{L}}{x^2+\left(\frac{\pi}{L}\right)^2}+\frac{1}{3}\left(\frac{\frac{3\pi}{L}}{x^2+\left(\frac{3\pi}{L}\right)^2}\right)+\frac{\frac{5\pi}{L}}{x^2+\left(\frac{5\pi}{L}\right)^2}+\dots\right]$$
$$=\frac{4}{\pi}\left[\frac{\frac{\pi}{L}}{x^2+\left(\frac{\pi}{L}\right)^2}+\frac{\frac{\pi}{L}}{x^2+\left(\frac{3\pi}{L}\right)^2}+\frac{\frac{\pi}{L}}{x^2+\left(\frac{5\pi}{L}\right)^2}\dots\right]$$
$$=\frac{4}{L}\left[\frac{1}{x^2+\left(\frac{\pi}{L}\right)^2}+\frac{1}{x^2+\left(\frac{3\pi}{L}\right)^2}+\frac{1}{x^2+\left(\frac{5\pi}{L}\right)^2}+\dots\right]$$
Hence, as there are no real singularities, therefore the number of poles are 0.

### Relevant Questions

In the arrangement shown in Figure P14.40, an object of mass,m = 2.0 kg, hangs from a cordaround a light pulley. The length of the cord between pointP and the pulley is L = 2.0 m.(a) When thevibrator is set to a frequency of 145Hz, a standing wave with six loops is formed. What must be thelinear mass density of the cord in kg/m? (b) How many loops (ifany) will result if m is changed to 2.88 kg?
(c) How many loops (if any) will result if m is changed to72.0 kg?
Write down the qualitative form of the inverse Laplace transform of the following function. For each question first write down the poles of the function , X(s)
a) $$X(s)=\frac{s+1}{(s+2)(s^2+2s+2)(s^2+4)}$$
b) $$X(s)=\frac{1}{(2s^2+8s+20)(s^2+2s+2)(s+8)}$$
c) $$X(s)=\frac{1}{s^2(s^2+2s+5)(s+3)}$$
An alpha particle (a He nucleus, containing two protons and two neutrons and having a mass of $$\displaystyle{6.64}\cdot{10}^{{-{27}}}$$ kg) traveling horizontally at 35.6 km/s enters a uniform, vertical, 1.10 T magnetic field.
A) What is the diameter of the path followed by this alpha particle?
B) What effect does the magnetic field have on the speed of the particle?
C) What are the magnitude of the acceleration of the alpha particle while it is in the magnetic field?
D) What are the direction of the acceleration of the alpha particle while it is in the magnetic field?
Use Laplace transform to solve the following initial-value problem
$$y"+2y'+y=0$$
$$y(0)=1, y'(0)=1$$
a) \displaystyle{e}^{{-{t}}}+{t}{e}^{{-{t}}}\)
b) \displaystyle{e}^{t}+{2}{t}{e}^{t}\)
c) \displaystyle{e}^{{-{t}}}+{2}{t}{e}^{t}\)
d) \displaystyle{e}^{{-{t}}}+{2}{t}{e}^{{-{t}}}\)
e) \displaystyle{2}{e}^{{-{t}}}+{2}{t}{e}^{{-{t}}}\)
f) Non of the above
Part II
29.[Poles] (a) For each of the pole diagrams below:
(i) Describe common features of all functions f(t) whose Laplace transforms have the given pole diagram.
(ii) Write down two examples of such f(t) and F(s).
The diagrams are: $$(1) {1,i,-i}. (2) {-1+4i,-1-4i}. (3) {-1}. (4)$$ The empty diagram.
(b) A mechanical system is discovered during an archaeological dig in Ethiopia. Rather than break it open, the investigators subjected it to a unit impulse. It was found that the motion of the system in response to the unit impulse is given by $$w(t) = u(t)e^{-\frac{t}{2}} \sin(\frac{3t}{2})$$
(i) What is the characteristic polynomial of the system? What is the transfer function W(s)?
(ii) Sketch the pole diagram of the system.
(ii) The team wants to transport this artifact to a museum. They know that vibrations from the truck that moves it result in vibrations of the system. They hope to avoid circular frequencies to which the system response has the greatest amplitude. What frequency should they avoid?

1. S1 and S2, shown above, are thin parallel slits in an opaqueplate. A plane wave of wavelength λ is incident from the leftmoving in a direction perpendicular to the plate. On a screenfar from the slits there are maximums and minimums in intensity atvarious angles measured from the center line. As the angle isincreased from zero, the first minimum occurs at 3 degrees. Thenext minimum occurs at an angle of-
A. 4.5 degrees
B. 6 degrees
C. 7.5 degrees
D. 9 degrees
E. 12 degrees
Use Laplace transform to find the solution of the IVP
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a) $$f{{\left({t}\right)}}={3}{e}^{{-{2}{t}}}$$
b)$$f{{\left({t}\right)}}={3}{e}^{{\frac{t}{{2}}}}$$
c)$$f{{\left({t}\right)}}={6}{e}^{{{2}{t}}} d) \(f{{\left({t}\right)}}={3}{e}^{{-\frac{t}{{2}}}}$$
The Laplace transform of the function $${\left({2}{t}-{3}\right)}{e}^{{\frac{{{t}+{2}}}{{3}}}}$$ is equal to:
a) $${e}^{{\frac{2}{{3}}}}{\left(\frac{2}{{\left({s}-\frac{1}{{3}}\right)}^{2}}-\frac{3}{{{s}-\frac{1}{{3}}}}\right)}$$
b) $${e}^{{\frac{2}{{3}}}}{\left(\frac{2}{{\left({s}-\frac{1}{{3}}\right)}^{2}}\cdot\frac{3}{{{s}-\frac{1}{{3}}}}\right)}$$
c) $${e}^{{\frac{2}{{3}}}}{\left(\frac{6}{{\left({s}-\frac{1}{{3}}\right)}^{2}}\right)}$$
d) $${\left(\frac{2}{{{\left({s}-\frac{1}{{3}}\right)}^{2}+\frac{2}{{3}}}}\right)}$$
Use properties of the Laplace transform to answer the following
(a) If $$f(t)=(t+5)^2+t^2e^{5t}$$, find the Laplace transform,$$L[f(t)] = F(s)$$.
(b) If $$f(t) = 2e^{-t}\cos(3t+\frac{\pi}{4})$$, find the Laplace transform, $$L[f(t)] = F(s)$$. HINT:
$$\cos(\alpha + \beta) = \cos(\alpha)\cos(\beta) - \sin(\alpha) \sin(\beta)$$
(c) If $$F(s) = \frac{7s^2-37s+64}{s(s^2-8s+16)}$$ find the inverse Laplace transform, $$L^{-1}|F(s)| = f(t)$$
(d) If $$F(s) = e^{-7s}(\frac{1}{s}+\frac{s}{s^2+1})$$ , find the inverse Laplace transform, $$L^{-1}[F(s)] = f(t)$$
$${y}\text{}-{4}{y}={e}^{{-{3}{t}}},{y}{\left({0}\right)}={0},{y}'{\left({0}\right)}={2}$$
a) $$\frac{14}{{20}}{e}^{{{2}{t}}}-\frac{5}{{30}}{e}^{{-{2}{t}}}-\frac{9}{{30}}{e}^{{-{6}{t}}}$$
b) $$\frac{11}{{20}}{e}^{{{2}{t}}}-\frac{51}{{20}}{e}^{{-{2}{t}}}-\frac{4}{{20}}{e}^{{-{3}{t}}}$$
c) $$\frac{14}{{15}}{e}^{{{2}{t}}}-\frac{5}{{10}}{e}^{{-{2}{t}}}-\frac{9}{{20}}{e}^{{-{3}{t}}}$$
d) $$\frac{14}{{20}}{e}^{{{2}{t}}}+\frac{5}{{20}}{e}^{{-{2}{t}}}-\frac{9}{{20}}{e}^{{-{3}{t}}}$$