Determine the moment of inertia of the Z section with respect to the x and y centroidal axes. 1 in. 4 in.- 4 in.- 1 in. T-4 in. -1 in. 8 in. X 1 in.

Draw the free-body diagram for the following problems. a) The beam in Prob. 510. b) The beam in Prob. 511. c) The beam in Prob. 512. d) The beam in Prob. 514.

The cylinder is confined by the brake, where s = 0.4. Determine the required compression in the spring in order to resist a torque of 800 N m on the cylinder. The spring has a stiffness of k = 3...

The car has a mass of 1.6 Mg and center of mass at G. If the coefficient of static friction between the shoulder of the road and the tires is s = 0.4 determine the greatest slope the shoulder can...

The 100-lb cylinder rests between the two inclined planes. When P = 15 lb, the cylinder is on the verge of impending motion. Determine the coefficient of static friction between the surfaces of...

Determine the location y of the centroid for the beams cross-sectional area. 15 mm 400 mm 15 mm - 300 mm -200 mm- 15 mm

Locate the centroid y of the area. 16 ft -y = (4-x) 4 ft- T 4 ft +

Determine the force P that must be applied perpendicular to the handle in order to hold the mechanism in equilibrium for any angle of rod CD. There is a couple moment M applied to the link BA. A...

When = 20 the 50-lb uniform block compresses the two vertical springs 4 in. If the uniform links AB and CD each weigh 10 lb, determine the magnitude of the applied couple moments M needed to...

The linkage is subjected to a force of P = 2 kN. Determine the angle for equilibrium. The spring is unstretched when = 0. Neglect the mass of the links. k = 15 kN/m A B -0.6 m- -0.6 m- D Prob. F11-3...

Determine the required magnitude of force P to maintain equilibrium of the linkage at = 60. Each link has a mass of 20 kg. A 1.5 m B Prob. F11-1 C 1.5 m

Determine the moment of inertia of the triangular area about the y axis. h y b - y = h (b x) X

Determine the centroid of the area. 2 m -1 m- -1 m- -y = 2x X

Locate the center of mass of the straight rod if its = x 0.5 (1+x) kg. mass per unit length is given by m = 0.5(1 + x 2 ) kg. -1m- Prob. F9-4 X

Determine the moment of inertia of the area about the y axis. y y = x 1 in. 1 in. X

Draw the free-body diagram for the following problems. a) The rod in Prob. 544. b) The hand truck and load when it is lifted in Prob. 545. c) The beam in Prob. 547. d) The boom frame in Prob. 552. k...

A common biological environment consists of large macro-ions with charge Q < 0 floating in a solution of point-like micro-ions with charge q > 0. Experiments show that N micro-ions adsorb onto the...

Suppose that the electric potential of a point charge at the origin were Evaluate a superposition integral to find inside and outside a spherical shell of radius R with uniform. Check the 0 limit....

Let A and B be vectors. Show that 1 1 = = ijk (A B)k + (A; B; + A;B;). 2 A; B; =

If a and b are constant vectors, (r) = (a r) (b r) is the electrostatic potential in some region of space. Find the electric field E = and then the charge density = 0 E associated with this...

Let b be a vector and n a unit vector. (a) Use the Levi-Civit`a symbol to prove that b = (b n) n + n (b n). (b) Interpret the decomposition in part (a) geometrically. (c) Let = a (b c) where...

Mimic the proof of Helmholtz theorem in the text and prove that y(r) = -V. 4 dr' V'y(r') [r - r' + V 1+1 7 Sa S ds'. (r') r-r'

Let E be a positive real number. Evaluate (a) (b) (c) 81(E)= f dk, 8(E k).

(a). Show that (r)/r = (r) when it appears as part of the integrand of a three-dimensional integral in spherical coordinates. Convince yourself that the test function (r) does not provide any...

Without using vector identities: (a) Use Stokes Theorem dS ( A) = ds A with A = c F where c is an arbitrary constant vector to establish the equality on the left side of (b) Confirm the equality...

Use the Levi-Civita symbol to prove that. (a) (f g) = f g + g f. (b) (f g) = f g gf. (c) (g r) = 2g + r g/r r( g).

A test function as part of the integrand is required to prove any delta function identity. With this in mind: (a) Prove that (ax) = 1/|a|(x), a 0. (b) Use the identity in part (a) to prove that (c)...

Use the method of Lagrange multipliers to show that, among all functions D(r) which satisfy D = f , the minimum (not merely the extremum) of: Occurs when D(r) = E(r). Assume that n D takes...

A spherical conductor of radius R 1 is surrounded by a polarizable medium which extends from R 1 to R 2 with dielectric constant . (a) The conductor has charge Q. Find E everywhere and confirm that...

Two concentric, spherical, conducting shells have radii R 2 > R 1 and charges q 2 and q 1 . The volume between the shells is filled with a linear dielectric with permittivity = 0 . Determine the...

A uniform electric field E 0 exists throughout a homogeneous dielectric with permittivity . What is the electric field inside a vacuum cavity cut out of the interior of the dielectric in the shape of...

(a) The entire volume between two concentric spherical shells is filled with a material with uniform polarization P. Find E(r) everywhere. (b) The entire volume inside a sphere of radius R is filled...

A semiconductor with permittivity occupies the space z 0. One dopes such a semiconductor by implanting neutral, foreign atoms with uniform density N D in the near-surface region 0 z d. Assume...

Confirm the assertion made in the text that the inverse relation between the matrix of capacitive coefficients and the matrix of potential coefficients implies the equivalence of these two...

A point dipole p is placed at r = r 0 outside a grounded conducting sphere of radius R. Use Greens reciprocity (and a comparison system with zero volume charge density) to find the charge drawn up...

A grounded metal plate is partially inserted into a parallel-plate capacitor with potential difference 2 1 > 0 as shown in the diagram below. Find the elements of the capacitance matrix. Assume...

Let d be the separation between two infinite, parallel, perfectly conducting plates. The lower surface of the upper plate has charge per unit area 1 > 0. The upper surface of the lower plate has...

Let V be a charge-free volume of space. Use an interior spherical multipole expansion to show that the average value of the electrostatic potential (r) over the surface of any spherical sub-volume...

The z-axis is the symmetry axis for an origin-centered ring with charge Q and radius a which lies in the x-y plane. A coplanar and concentric ring with radius b > a has charge Q. Calculate the lowest...

A point charge q lies a distancer > R from the center of an uncharged, conducting sphere of radius R. Express the induced surface charge density in the form where is the polar angle measured from a...

An asymptotic (long-distance) electrostatic potential has the form (a) Use a traceless Cartesian multipole expansion to show that no localized charge distribution exists which can produce an...

Use the electric stress tensor formalism to prove that no isolated charge distribution (r) can exert a net force on itself. Distinguish the cases when (r) has a net charge and when it does not.

Find the primitive, Cartesian monopole, dipole, and quadrupole moments for each of the following charge distributions. Use the geometrical center of each as the origin. (a) Two charges +q at two...

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...

Consider a charge distribution (r) in rigid, uniform motion with velocity . (a) Show that the magnetic field produced by this system is B(r) = (/c 2 ) E, where E(r) is the electric field produced by...

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.

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,...

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...

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...

When coal is burned to form synthesis gas (syngas), which contains mostly CO, H 2 , H 2 S, and CO 2 , the H 2 S must be removed. The gas is called syngas because H 2 and CO are the building blocks...

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.