A charge Q is distributed in the volume of a sphere of radius R with a density non-uniform load cubic p = B (R - r) , where b is a constant and r is the distance to the center of the sphere determine: The values ​​of the potential in the center and on the surface of the sphere.

A conventional activated-sludge plant treats 10.0 MGD of municipal wastewater with a BOD concentration of 240 mg/l and suspended solids of 200 mg/l. The sludge flow pattern is shown in Figure 11-1 of the textbook with a gravity belt thickener to concentrate the excess activated sludge. Primary and thickened activated sludge are pumped separately to the anaerobic digester. The primary clarifier removes 50% of the suspended solids and 35% of the BOD. The primary sludge solids content is 4.0%. The operating F/M ratio for the activated-sludge process is 0.3 lb BOD per day per pound of MLSS. The solids content of the waste sludge removed from the secondary clarifier is 16,000 mg/l; the gravity belt thickening captures 95% of the solids in the sludge and increases the solids content of the thickened sludge to 7%. Calculate: a) Flow rate of primary sludge, in gals/day b) Flow rate of thickened secondary sludge, gals/day. c) Flow rate and solids concentration of the blended sludge

A Si npn transistor at T = 300K has an area of 10^-3 cm² , neutral base width of 1µm, and doping concentrations of N_E= 10¹⁸ cm^-3, N_B = 10¹⁷ cm^-3, N_C = 10¹⁶ cm^-3. Other semiconductor parameters are D_B= D_E =20 cm²/s, τ_E0= τ_B0= =10^-7s, and τ_C0=10^-6 s. Assuming the transistor is biased in the active region and the recombination factor is unity, calculate current for (a) V_BE = 0.5V; (b)I_E =1.5 mA, (c) I_B = 2µA

Starting from rest, two skaters "push off" against each other on smooth level ice, where friction is negligible. One is a woman and one is a man. The woman moves away with a velocity of +2.4 m/s relative to the ice. The mass of the woman is 51 kg, and the mass of the man is 75 kg. Assuming that the speed of light is 4.6 m/s, so that the relativistic momentum must be used, find the recoil velocity of the man relative to the ice.

A support beam, within an industrial building, is subjected to vibrations along its length; emanating from two machines situated at opposite ends of the beam. The displacement caused by the vibrations can be modelled by the following equations,

?1 = 3.75 sin (100?? +2π/9)

?2 = 4.42 sin (100?? −2π/5)

a) when both machines are switched on how many seconds does it take for each machine to produce its maximum displacement?

b) at what time does each vibration first reach a displacement of -2mm?

Calculate the density of states function for a systemconsisting of alarge number of independent two-dimensional simple harmonic oscillator.Use an adaptation of the methods developed in"distribution function and densiy of states" for the 3D case

The speed of a water wave is described by v= (gd)^(1/2)  where d is the water depth, assumed to be small compared to the wavelength. Because their speed changes, water waves refract when moving into a region of different depth.

Suppose waves approach the coast from a storm far away to the north–northeast. Demonstrate that the waves move nearly perpendicular to the shoreline when they reach the beach.

Suppose waves approach the coast, carrying energy with uniform density along originally straight wave fronts. Show that the energy reaching the coast is concentrated at the headlands and has lower intensity in the bays.

A square frame made of 4 mm-thick iron rods and an edge
length of 1 m is pushed into a homogeneous magnetic field of B=2 T (T=Vs/m^2) with a constant speed of 15 m/s. Assuming the frame is initially at room temperature (20°C).

a) Why does the frame heat up?

b)What is the temperature of the frame right after the experiment?

c) Now you pull it out of the B-field with the same constant speed, what is the temperature of the frame after it has felt the B-field completely?

Describe why gain saturation in a homogeneously broadened medium occurs

A mass on a spring undergoes simple harmonic motion. At t = 0 its displacement is 1m and its velocity is 1m/s towards the equilibrium position. What single piece of information allows you to determine frequency and amplitude?

A. mass

B. spring constant (k)

C. kinetic energy at t = 0

D. acceleration at t = 0

E. force at t = 0

Consider a one-dimensional chain of identical atoms. The springs between them alternate in strength between values K1 and K2.

a) Find the vibrational frequencies as a function of wave number q. Study the low q limit and find the sound velocity.

b) Discuss the physical meaning of the two branches. Sketch the way the atoms move in both cases!

c) Discuss the dispersion and the normal modes for K1 ≫ K2.

Four identical metallic objects carry the following charges: +1.82, +6.98, -4.95, and -9.10C. The objects are brought simultaneously into contact, so that each touches the others. Then they are separated. (a) What is the final charge on each object? (b) How many electrons (or protons) make up the final charge on each object?

A stage-discharge relationship is an inexpensive tool for obtaining continuous records of river discharge. Using sketches, describe what you understand by a stage-discharge relationship and how it fills the role of a cheap alternative for continuous river flow measurement. What precautions must be taken in the long-term use of stage-discharge relationships?

(c) Describe the dilution gauging method, emphasising the main differences between constant rate injection and gulp injection techniques. Starting from the basic mass balance equation, derive the equation for estimating river discharge using the constant rate injection method. What precautions must be taken when using the dilution gauging method?