Determine the moment of inertia of the area about the x axis. -3 in.3 in. 2 in. 6 in. 4 in. X

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 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...

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...

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...

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...

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...

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...

Molecules adsorbed on the surface of a solid crystal surface at low temperature typically arrange themselves into a periodic arrangement, e.g., one molecule lies at the center of each a a square of...

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...

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 ...

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'...

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...

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

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...

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...

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...

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...

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,...

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. Example 13-1: Batch Reactor with an Exothermic Reaction Wolfram 1. Adiabatic Case: Use Wolfram to see whether you can find a trajectory that is ready to ignite and whose trajectory looks like a...

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...

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?

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...

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?

When a 120-lb woman stands on a snow-covered trail, she sinks slightly into the snow because the compressive stress between her ski boots and the snow is larger than the snow can support without...

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...

(a) By using either the vector algebra or polygon methods for finding a resultant, determine the magnitude of F that will cause the net effect of the three forces to act vertically. (b) For that...

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...

The steel bolt and anchor assembly is used to reinforce the roof of a passageway in an underground coal mine (Figure P5.13). In installation, the bolt is tensioned to 5000 lb. Calculate the stress,...

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...

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 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...

(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 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...

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?...

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 =...

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...

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...