The electrical power generated by a wind turbine can depend only on L, the length of the turbine’s blades, ρ the mass density of the air, and v the speed of the air. Use dimensional analysis to show that the power created by the turbine must be proportional to the cube of the air speed, v. If the air speed doubles the power generated will increase by nearly an order of magnitude! Note that power in the MKS system is measured in Watts where 1[W] = 1 [J/ s] = 1 [kg · m^2/ s^3] .

(a) Once an ideal inductor is connected to a 220 V, 50 Hz a.c supply and later the same inductor is connected to 220 V, 100 Hz a.c supply. What happens to the current flowing through it in both the cases? Will the current be the same or different? Give brief reason (2marks)

(b) A current of 12A flows in a when connected to a 240V , 50Hz AC mains and 2A from a 12V DC mains. Find the values of (i) the resistance (ii) impedance (iii) inductance of the coil         (3marks)

1. Let V = { S, A, B, a, b, λ} and T = { a, b }, Find the languages generated by the grammar G = ( V, T, S, P } when the set of productions consists of:
   1. S → AB, A → aba, B → bab.
   2. S → AB, S → bA, A → bb, B → aa.
   3. S → AB, S → AA, A → Ab, A → a, B → b.
   4. S → A, S → B, A → abA, A → a, B → baB, B → b.
   5. S → AB, A → aBb, B → bAa, A → λ, B → λ.

Chem Lab/Biology Question

You need to prepare 350ml of an extraction buffer with the following composition: 125 mM NaCl (MW=58.44), 100mM Tris base (MW=121.1), and 3% (w/v) sodium dodecyl sulfate. You have access to water, powdered Tris base, 6 M NaCl solution, and 40% (w/v) solution of sodium dodecyl sulfate. How much of each component is needed (you may assume that solids contribute negligible volume to a solution)?

Water in a piston-cylinder assembly is at 1 MPa, 500◦C. There are two stops, a lower one at which Vmin = 1 m3 and an upper one at Vmax = 3 m3. The piston is loaded with a mass and outside atmosphere such that it floats when the pressure is 500 kPa. This setup is now cooled to 100◦C by rejecting heat to the surroundings at 20◦C. a. Show the path of the process on T-V and P-V diagrams. b. Determine the heat transferred and work done during this process.

Let ? ∈ {1, 2} and ? ∈ {3, 4} be independent random variables with PMF-s:

?_x(1)= 1/2 ?_x(2)= 1/2 ?_Y(3)= 1/3 ?_Y(4)= 2/3

(a) Write down the joint PMF

(b) Calculate ?(? + ? ≤ 5) and ? (? − ? ≥ 2)

(c) Calculate ?(?? ), ?(?²? ),Calculate E((X²+1)/(Y-2))

(d) Calculate the C??(?, ? ), C??(1 − ?, 3? + 2) and V??(2? − ? )

(e) Calculate C??(??, ?), C??(??, ? + ? ) and V?? (?/Y)

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x +2y +z, 2x +2y +3z +w, x +4y +2w)

a) Find the dimension and basis for Im T (the image of T)

b) Find the dimension and basis for Ker ( the Kernel of T)

c) Does the vector v= (2,3,5) belong to Im T? Justify the answer.

d) Does the vector v= (12,-3,-6,0) belong to Ker? Justify the answer.

Consider four 60·nC charges placed at the corners of a square, which is 3.6·m on a side. The charges on the bottom are both positive, and the charges on the top are both negative.

(a) Find the electric field at the center of the square, in units of V/m. vector E = 0 i hat + 235.7 j

(b)Find the electric field at the midpoint of the right side, in units of V/m. vector E = _ i hat + _ j

Please help find the answer to part B.

A. Show that the ideal gases following PV=nRT satisfy (dp/dv)(dv/dt)(dt/dp)=-1.

B. Show that van see waals fluids following P=(RT/V-b) - a/V^2 satisfy (dp/dv)(dv/dt)(dt/dp)=-1.

C. Use the fundamental relation, dH=TdS + VdP and Maxwell equations to show that (dH^ig/dV) = 0, which means that enthrall of ideal gases us independent of volume.

1) You have three vectors with (∣v⃗∣,θ)(∣v∣,θ) pairs of (11,75∘)(11,75∘), (8,50∘)(8,50∘), and (10,−100∘)(10,−100∘).

If you wanted to add a fourth vector to those first three such that the sum of all four vectors were 0, what would that fourth vector be? Give your answer as a (vx,vy)(vx​,vy​) pair. Give your answer to 1 decimal point. Then show it as a (∣v⃗∣,θ)(∣v∣,θ) pair.