Question 106 - In a wavefunction, let P(x) = A cos kx + B sin kx. By using the boundary conditions of x = 0 and x = l, where P(0) = P(l) = 0, prove by mathematical calculation that P(x) = B sin (npx / l) where p = 22 / 7 approximately, n is a rounded number. A, B and k are constants.
Answer 106 - When x = 0, P(0) = A cos (k x 0) + B sin (k x 0) = A = 0. When x = l, P(l) = A cos (k x l) + B sin (k x l) = 0 = B sin (k x l), then k x l = arc sin 0 = np, k = np / l. Finally P(x) = B sin kx = B sin (npx / l) as proven. The answer is given by Kang Chuen Tat; PO Box 6263, Dandenong, Victoria VIC 3175, Australia; SMS +61405421706; chuentat@hotmail.com; http://kangchuentat.wordpress.com.
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Question 47 - In a cylinder with a hollow, let a is outside radius and b is the inside radius. In a steady state temperature distribution with no heat generation, the differential equation is (d / dr) (r dT / dr) = 0 where r is for radius and T is for temperature. (a) Integrate the heat equation above into T(r) in term of r. (b) At r = a, T = c; at r = b, T = d. Find the heat equation of T(r) in term of r, a, b, c, d.
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BIOPROCESS ENGINEERING - EXAMPLE 14.1 : In differential centrifugation of cells with diameter D in centimeter, the square of D is given by D x D = [18n ln (RF / RI) ] / [ (RP - RFF) Wt ] where n is the fluid viscosity (poise), RF is the final radius of rotation (cm), RI is the initial radius of rotation (cm), RP is cell density (g / ml), RFF is the fluid density (g/ml), W the square for the rotational velocity in (radians / s) (radians / s), t is the time required to sediment from RI to RF (s). Derive an equation for W as a function for D, n, RF, RI, RP, RFF and t, with the stated units above, in radian & degree.
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QUANTUM BIOLOGY - EXAMPLE 33.4 : (a) According to Landauer (1986), the capacity of human memory is approximately X bits. Assume that a human retains 2 bits / second of visual, verbal, tactile and musical memory, find the value of X if a human lifetime is approximately 2.5 billion seconds. (b) The total power consumption of the human brain is about 25 Watts. The bread of 100 grams will produce 1000 kilojoules of energy. How much bread is needed to run a human brain for 1 day?
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