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A surprisingly realistic microscopic model for the charge density of a semi-infinite metal (with z = 0 as its macroscopic surface) consists of a positive charge distribution And a negative charge... An infinitely long conducting cylinder (radius a) oriented along the z-axis is exposed to a uniform electric field E 0 y. (a) Consider the conformal map g(w) = w + a 2 /w, where g = u + iv and w = x... A set of known constants n parameterizes the potential in a volume r < a as Let z point along = 0 and insert a solid conducting sphere of radius R < a at the origin. Show that the force exerted on... A conducting sphere with radius R and charge Q sits at the origin of coordinates. The space outside the sphere above the z = 0 plane has dielectric constant 1 . The space outside the sphere below... Two semi-infinite, hollow cylinders of radius R are coaxial with the z-axis. Apart from an insulating ring of thickness d 0, the two cylinders abut one another at z = 0 and are held at potentials V... The figure below shows an infinitely long cylindrical shell from which a finite angular range has been removed. Let the shell be a conductor raised to a potential corresponding to a charge per unit... The square region defined bya x a anda y a in the z = 0 plane is a conductor held at potential = V . The rest of the z = 0 plane is a conductor held at potential = 0. The plane z = d is also a... Determine the moment of inertia of the triangular area about the x axis. h y -b - y = (b x) x Determine the x, y, z components of force and moment at point C in the pipe assembly. Neglect the weight of the pipe. The load acting at (0, 3.5 ft, 3 ft) is F 1 = 5-24i - 10k6 lb and M = {-30k} lb ... The 0.2-lb ball bearing travels over the edge A with a velocity of v A = 3 ft/s. Determine the speed at which it rebounds from the smooth inclined plane at B.Take e = 0.8. V = 3 ft/s 45 B The shaft is constructed from a rod which has a mass per unit of 2 kg/m. Determine the x, y, z components of reaction at the bearings A and B if at the instant shown the shaft spins freely and has an... Show that can be used as the complex potential for an array of equally spaced, parallel, charged lines in the y = 0 plane. Let n be an integer and let x = na and x = (n + 1/2 )a be the positions of... Positive charges Q and Q' are placed on opposite sides of a grounded sphere of radius R at distances of 2R and 4R, respectively, from the sphere center. Show that Q' is repelled from the sphere if Q'... The diagram below shows a rod of length L and net charge Q (distributed uniformly over its length) oriented parallel to a grounded infinite conducting plane at the distance d from the plane. (a)... (a) Let (,) be a solution of Laplaces equation in a cylindrical region < R. Show that the function (,) = (R 2 /,) is a solution of Laplaces equation in the region > R. (b) Show that a suitable... Let b be the perpendicular distance between an infinite line with uniform charge per unit length and the center of an infinite conducting cylinder with radius R = b/2. (a) Show that the charge... A potential difference V drives a steady current I through an ohmic wire with conductivity and a constant circular cross section. One portion of the wire has the shape of a circular arc with inner... The diagram below shows that the resistance between the terminals A and B is determined by a motif of three resistors R 2 R 1 R 2 repeated an infinite number of times. Determine the ratio R 1 /R 2... The text describes a Helmholtz coil as two parallel, coaxial, and circular current loops of radius R separated by a distance R. Each loop carries a current I in the same direction. (a) Use the... A current distribution produces the vector potential What is the magnetic moment associated with this current distribution? 0 A(r, 0, 0) = o Ao sin $ - exp(-r). 4 r (a) Find the vector potential inside and outside a solenoid that generates a magnetic field B = Bz inside an infinite cylinder of radius R. Work in the Coulomb gauge. (b) The A haronov-Bohm effect... Show that a current density with vector potential A(r) = f (r)r has zero magnetic dipole moment. It is true (but not obvious) which any vector field V(r) which satisfies V(r) = 0 can be written uniquely in the form where L = ir is the angular momentum operator and (r) and (r) are scalar... Determine the force in member BE. AE is constant. 3 m B E -3 m Prob. 9-29 D C 12 kN 3 m The correct form of the interaction energy between two current carrying circuits was much debated in the 1870s by Maxwell and the German theoretical physicists Carl Neumann, Wilhelm Weber, and... Estimate the liquid density at their boiling points for the following: 1. 2-butanol; 2. Methyl chloride; 3. Methyl ethyl ketone; 4. Aniline; 5. Nitrobenzene. The beam shown in Figure P567 carries a repeated, reversed load of 400 N applied at section C. Compute the resulting design factor if the beam is made from SAE 1340 OQT 1300. Data in Figure P567. 12... For the complex reaction with stoichiometry A + 3B 2R + S and with second-order rate expression are the reaction rates related as follows: r A , = r B , = r R ,? If the rates are not so related,... Calculate the source pressure required to deliver 13 x 10 6 standard cubic feet per day (SCFD) of CO to a customer 60 mi. away. By contract the pressure at the customer's site must be at least 300 lb... In a manufacturing plant, 100 mm by 40 mm thin rectangular electronic devices are assembled in mass quantity. The top surface of the electronic device is made of aluminum and is attached with a HS... If U is considered a function of T and P, the natural heat capacity is neither C V nor C P , but rather the derivative (U/T) P . Develop the following connections between (U/T) P , C P , and C V : To... In the steady-state process shown in the following fi gure, air fl ows through a turbine. Does the temperature increase, stay the same, or decrease? Explain. Air in P T Turbine Air out P T Derive the expression for the shape-efficiency factor \(\phi_{B}^{e}\) for stiffness-limited design for a square box section of wall thickness \(t\), and height and width \(h_{1}=10 t\) bent about... A triple effect standard vertical tube evaporator each effect of which has 140 m 2 of heating surface is to be used to concentrate from 5 % solids to 40% solids a solution possessing a negligible... Using tabulated experimental data from the literature for either the excess molar volume or excess molar enthalpy of a system of your choice, provide the following information. A. A plot of the... a. Determine the allowable torque, \(T\), if the crack length for the shaft in Figure 9.8 is \(2 a=10 \mathrm{~mm}\). Use the same dimensions and fracture toughness values that were given in Problem... Write the names and abbreviations for each of the prefixes from 10 -24 to 10 24 . Which abbreviations are capitalized? Compute the molecular mass and its standard uncertainty for NH 3 . What is the percent relative uncertainty in molecular mass? Find [La 3+ ] in the solution when excess solid lanthanum iodate, La(IO 3 ) 3 , is stirred with 0.050 M LiIO 3 until the system reaches equilibrium. Assume that IO - 3 from La(IO 3 ) 3 is negligible... Find the concentration of Cu + in equilibrium with CuBr(s) and 0.10 M Br - . Using activity coefficients, find the pH and fraction of dissociation of 50.0 mM hydroxybenzene (phenol) in 0.050 M LiBr. Take the size of C 6 H 5 O - to be 600 pm. Which National Institute of Standards and Technology buffer(s) would you use to calibrate an electrode for pH measurements in the range 34? What makes a liquid-based ion-selective electrode specific for one analyte? What is an absorption spectrum? Why is it most accurate to measure absorbances in the range A = 0.32? A spectrum has a signal-to-noise ratio of 8/1. How many spectra must be averaged to increase the signal-to-noise ratio to 20/1? Express your weight in the units of pounds and newtons, and your mass in the units of slugs and kilograms. Express your height in the units of inches, feet, and meters. a. With what initial speed must you throw a ball vertically for it to reach a height of 20 ft? The ball weighs 1 lb. How does your answer change if the ball weighs 2 lb? b. Suppose you want to throw... A passenger automobile is advertised as having a fuel economy rating of 29 mi/gal for highway driving. Express the rating in the units of km/L. (a) How many horsepower does a 100-W household lightbulb consume? (b) How many kW does a 5-hp lawn mower engine produce? Grade 1020 steel has a yield strength of 42 ksi and an elastic modulus of 30 Mpsi. Another grade of steel has a yield strength of 132 ksi. What is its elastic modulus? Find a real physical example of a mechanical structure or machine that has a moment acting on it. (a) Make a clear, labeled drawing of the situation. (b) Estimate the dimensions, and the magnitudes... Cable AB on the boom truck is hoisting the 2500-lb section of precast concrete (Figure P4.18). A second cable is under tension P, and workers use it to pull and adjust the position of the concrete... The pressure at the bottom of an 18-ft-deep storage tank for gasoline is how much greater than at the top? Express your answer in the units of psi. Blood pressure is conventionally measured in the dimensions of millimeters in a column of mercury, and the readings are expressed as two numbers, for example, 120 and 80. The first number is called... As a machinist presses the handles of the compound-action bolt cutters, link AB carries a 7.5-kN force (Figure P5.6). If the link has a 14 x 4-mm rectangular cross section, calculate the tensile... In a machine shop, the band saw blade is cutting through a work piece that is fed between the two guide blocks B (Figure P5.8). To what tension P should the blade be set if its stress is to be = ksi... Determine the elastic modulus and the yield strength for the material having the stressstrain curve shown in Figure P5.11. Use the 0.2% offset method. Figure P5.11 1600 1400 1200 1000 800 600 400... Make an order-of-magnitude estimate for a safety floatation device that can be used by children who are playing in a swimming pool. The design concept is for one inflatable, annular, plastic balloon... Scuba divers carry ballast weights to have neutral buoyancy. At that condition, the buoyancy force on the diver exactly balances weight, and there is no tendency either to float toward the surface or... For the system in Problem P5.15, assume that two 180-lb workers are going to be standing on the system to repair a bridge, one on each platform. The following design requirement must be met: The... The small steel plate is connected to the right angle bracket by a 10-mm diameter bolt (Figure P5.20). Determine the tensile stress at point A in the plate and the shear stress in the bolt. Figure... A detailed view is shown for the connection B in the concrete trough from Problem P4.17 in Chapter 4 (Figure P5.22). Determine the shear stress acting in the shackles 3/8 -in. bolt. Figure P5.22... The average velocity of blood flowing in a certain 4-mm-diameter artery in the human body is 0.28 m/s. Calculate the Reynolds number and determine whether the flow is laminar or turbulent. The... (a) For a 1.25-in.-diameter pipe, what is the maximum volumetric flow rate at which water can be pumped and the flow will remain laminar? Express your result in the dimensions of gallons per minute.... A small water droplet in a mist of air is approximated as being a sphere of diameter 1.5 mil. Calculate the terminal velocity as it falls through still air to the ground. Is it reasonable to neglect... A 6-m-high, 4-m-wide rectangular gate is located at the end of an open freshwater tank (Figure P6.6). The gate is hinged at the top and held in place by a force F. From Equation (6.4), the pressure... For the system in Problem P6.6, suppose your design requirement was to minimize the force necessary to hold the gate closed. Would you rather put the hinge at the top of the gate or at the bottom?... A 175-lb skydiver reaches a terminal velocity of 150 mph during free fall. If the frontal area of the diver is 8 ft 2 , what are: (a) The magnitude of the drag force acting on the skydiver? (b) The... A 14-mm-diameter sphere is dropped into a beaker of SAE 30 oil. Over a portion of its downward motion, the sphere is measured to fall at 2 m/s. In the units of newtons, what is the drag force acting... A submarine releases a spherical flotation buoy containing a radio beacon. The buoy has a diameter of 1 ft and weighs 22 lb. The coefficient of drag for the submerged buoy is C D = 0.45. At what... An ancient kings supposedly golden crown had a mass of 3 kg, but it was actually made by a dishonest metal smith from an equal mix of gold (1.93 10 4 kg/m 3 ) and silver (1.06 10 4 kg/m 3 ). (a)... For the parabolic pressure profile in Equation (6.7), show that the average flow velocity is half the maximum value [Equation (6.11)]. Equation 6.7 Equation 6.11 (1 (5) (Special case: Re < 2000) v =... Submarines dive by opening vents that allow air to escape from ballast tanks and water to flow in and fill them. In addition, diving planes located at the bow are angled downward to help push the... A sprinter runs using a force of 200 N and a power output of 600 W. Calculate how many minutes it takes for the runner to run 1 km. The volume of a parallelepiped can be computed from |A - (B C)|, where A, B, and C define three sides of the parallelepiped (see Figure P34). Compute the volume of a parallelepiped defined by A +... Aparticular circle is centered at the origin and has a radius of 5. Use MuPAD to find the equation of the line that is tangent to the circle at the point x = 3, y = 4. In a sandy soil (' = 35; = 18.0 kN/m 3 , and sat = 20.0 kN/m 3 ), the water table is located at a depth of 6 m below the ground level. How deep would you drive a sheet pile to carry out an... Use MuPAD to prove the following identities: a. sin 2 x = cos 2 x = 1 b. sin(x + y) = sin x cosy + cosx siny c. sin 2 x = 2 sin x cosx d. cosh 2 x - sinh 2 x = 1 A soil profile is shown in Figure P3.3 along with the standard penetration numbers in the clay layer. Use Eqs. (3.8b) and (3.9) to determine the variation of c u and OCR with depth. What is the... A standard penetration test was carried out in a normally consolidated sand at 7.62 m depth where the N 60 was determined to be 28. The unit weight of the sand is 17.29 kN/m 3 , and the grain-size... The 20-N horizontal force acts on the handle of the socket wrench. What is the moment of this force about point B. Specify the coordinate direction angles , , of the moment axis. 20 N -200 mm- B A... Use the general bearing capacity equation [Eq. (6.28)] to solve the following: a. Problem 6.1a b. Problem 6.1b c. Problem 6.1c Eq. (6.28) qu = c'N,FeFcaFeci + qNFqsFqaFqi + >yBN,F,FyaFyi +... Replace the loading system acting on the post by an equivalent resultant force and couple moment at point A. 650 N 500 N 30 300 N 1500 N m 60 B -3 m 5 m 2 m A 1500 kN load was applied on two 20 m long and 500 mm diameter piles that were instrumented for measuring the load variation with depth. a. The variation of frictional resistance per unit area, f... Redo Problem 12.3 using Coyle and Castellos methods for estimating both Q p and Q s . Problem 12.3 A 500 mm diameter and 20 m long concrete pile is driven into a sand where = 18.5 kN/m 3 and ' = 32.... Consider a 500 mm diameter pile having a length of 18 m in a clay. Given: = 20.0 kN/m 3 and c u = 60 kN/m 2 . a. Determine the maximum allowable load (Qall) with FS = 3. Use the a method and Table... The form is used to cast a concrete wall having a width of 5 m. Determine the equivalent resultant force the wet concrete exerts on the form AB if the pressure distribution due to the concrete can be... Consider a continuous flight auger pile in a sandy soil deposit 10 m long with a diameter of 0.45 m. Following is the variation of standard penetration resistance values (N 60 ) with depth. Depth (m)... Determine the magnitude of force at the pin A and in the cable BC needed to support the 500-lb load. Neglect the weight of the boom AB. 8 ft 122 35 The average values of the undrained shear strength (cu) with depth are given as follows: Depth (m) . c u (kN/m 2 ) 03 .. 20 38 .. 35 812 . 80 1216 . 95 If a CFA pile having a diameter of 0.6 m and... A steel pile (H-section; HP 310 125; see Table 12.1a) is driven into a layer of sandstone. The length of the pile is 25 m. Following are the properties of the sandstone: unconfined compression... Determine the force in each member of the truss and state if the members are in tension or compression. Set P1 = 10 kN, P2 = 8 kN. F E 2 m D. A B 2 m P2 Figure P13.1 shows a drilled shaft in sand. Determine the maximum load that can be allowed on it. Assume ' = 0.6' and allow a factor of safety of 4. Use Berezantzev et al.s Eq. (13.18) for computing... A straight drilled shaft 1.0 m in diameter and 15.0 m long is installed in a sand where ' = 34 and = 18.0 kN/m 3 . a. Determine the load-carrying capacity at the base using the Chen and Kulhawy... The toggle clamp is subjected to a force F at the handle. Determine the vertical clamping force acting at E. The top 5 m in a site consists of a clay with c u = 80 kN/m 2 . This is underlain by a thick clay deposit with c u = 150 kN/m 2 . A 1.0 m diameter and 15 m long straight drilled shaft is constructed... Figure P16.8 shows a smooth vertical wall retaining a sandy backfill underlain by clay. Assuming that the entire soil is in the active state, determine the magnitude and location of the resultant... In Problem 17.1, if there is a possibility that the soil in front of the wall will be removed sometime later, neglect the passive resistance and compute the factors of safety. Problem 17.1 Figure...
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