Perturbation theory and calculus of variations (840G1)

 Sheet 1. The Pi Theorem

微扰理论代写 Submit your solutions to all assessed exercises by 14:00pm on Thurs- day 21st February 2019 (Week 3). For late…

Submit your solutions to all assessed exercises by 14:00pm on Thurs- day 21st February 2019 (Week 3). For late submissions the following rules apply: 5 marks penalty for being up to 24h late and a score of 0 (zero) if more than 24h late. Your work should be submitted directly to the School Mathematical and Physical Sciences, Pevensey II

  1. AssessedExercise: A pendulum executing small vibrations has period p and length l, and m is the mass of the bob.

(a) Canp depend only on l and m? Justify your 微扰理论代写

(b) Ifp only depends on l and acceleration g due to gravity, then show that the period p can be written as

  1. Assessed Exercise: While biking, the force of air resistance F (masstimes length, per time-squared) appears to be positively related to the speed v (length per time) at which the bike travels. And the cross-sectional area A (length-squared) of bike and biker. However, the force involves mass and hence cannot depend on v. And A  Let the force also depend on the air density ρ (mass per length-cubed). Hence, the force F can be expressed as some multiplicative combination of v, A and ρ, that is

F kvxAyρz.

Find xyz so that the above equation is dimensionally correct.

  1. Assessed Exercise: Suppose that for some physical phenomenon pres-sure (P ), length (l), mass (m), time (t) and density (ρ) are related via the following law
微扰理论代写
微扰理论代写

Is this law unit-free? 微扰理论代写

微扰理论代写
微扰理论代写
  1. Assessed Exercise:In an indentation experiment, a slab of metal of thickness h is subjected to a constant pressure P on its upper surface by a cylinder of radius a. The technician then measures the vertical dis- placement U of the indentation. The displacement also depends upon two material properties, Poisson’s ratio w, which is dimensionless. And the Lam´e constant µ, which has dimensions M/L3T 2, where M  is mass, L is length and T is time. Show that there is an equivalent physical law of the form
微扰理论代写
微扰理论代写

Hint:- It is enough to show that the solution space of AX = 0 is 3- dimensional and that the vectors X1X2 and X3 corresponding to  ,

  and  are basis vectors. Why is this enough? ]

  1. Usedimensional analysis to prove Pythagoras’s

You may assume that 微扰理论代写

  • in a right-angled triangle, the area A is determined by the length c of the hypotenuse. And the magnitude φ of the smallest internal angle

and

  • the perpendicular dropped from the vertex opposite the hypotenuse to the hypotenuse divides the triangle into two right-angled triangles, each of which is similar to the given triangle.
 
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