Serial of year 24

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Tasks

1. Series 24. Year - S. complex warm-up

 

  • Convince yourselves that the nth roots of a complex number of modulus one lie on a regular $n-gon$ and solve the Bombelli equation $x^{3}-15x-4$ = 0. (see the text for hints)
  • Express the identities concerning sin(α+β) and cos(α+β) using the complex exponential.
  • Show that we were allowed to neglect the higher powers in deriving the Bernoulli limit, i.e. show that it was legitimate to add the o(1/$N)$ term inside the parentheses.
  • Use the little-o notation to solve the problem of small oscillations around equilibrium point in Yukawa potential $V$ = $k \exp(x/λ) /$ $x$.
  • Prove that the Chebyshev polynomials cos($n$ arccos $x)$ are really polynomials.

Hint: Let's have a unit complex number $z$ with real part $x$. Then, the expression is equal to the real part of $z^{n}$.

Jakub Michálek a Lukáš Ledvina

5. Series 24. Year - S. aviation

 

  • How would you draw (using only drawing-compass and a ruler) the Zhukovsky profil?
  • Draw the streamlines around the Zhukovsky profile. Choose the parameters $d/l$ and $m/l$ so that they have real world justification.
  • What is the lifting force acting on a square board? On a semicircular board?
  • Draw the profile of a wing that corresponds to the Kármán–Trefftz transform.

Jakub

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