The stagnation temperature on the Apollo vehicle at Mach 36 as it entered the atmosphere was 11,000 K, a much different value than predicted in Problem 8.17 for the case of a calorically perfect gas...

A unity feedback system has the loop transfer function Determine the relationship between the steady-state error to a ramp input and the gain K and system parameter b. For what values of K and b can...

Extreme temperature changes result in many failures of electronic circuits. Temperature control feedback systems reduce the change of temperature by using a heater to overcome outdoor low...

A T-frame structure is composed of prismatic beam ABC and non prismatic column DBF that are joined at B by a frictionless pin connection. The beam has sliding support at A and the column is pin...

A simply supported beam is subjected to a linearly varying distributed load q(x) 0 maximum intensity q 0 at B. The beam has a length L = 4 m and a rectangular cross-section with a width of 200 mm and...

Determine the magnitude of the resultant force F R = F 1 + F 2 and its orientation , measured counter clockwise from the positive x axis. F3 = 250 N F = 360 N 30% 45% 30 F = 400 N X

Determine the tension developed in cables OD and OB and the strut OC, required to support the 50-kg crate.The spring OA has an unstretched length of 0.8 m and a stiffness The force in the strut acts...

Determine the tension in the cables in order to support the 100-kg crate. x- C B 2 m N 2.5 m m D 2 m 1 m -y

The 25-kg flowerpot is supported at A by the three cords. Determine the force acting in each cord. x D N 60 30 30 B 45

If the bucket and its contents have a mass of 10 kg and are suspended from the cable by means of a small pulley at C, determine the location x of the pulley for equilibrium. The cable is 6 m long. 4...

Each cord can sustain a maximum tension of 500 N. Determine the largest mass of pipe that can be supported. FM E H B 60 1 Y H D E 25

If the mass of the block at A is 20 kg, determine the mass of the block at B and at C for equilibrium. Also, determine the angles u and f. A D 3 m B E -4 m C 3 m

The 30-kg pipe is supported at A by a system of five cords. Determine the force in each cord for equilibrium. G B 60 A H D E

Let P = 2a x 4a y + a z and Q = a x + 2a y . Find R which has magnitude 4 and is perpendicular to both P and Q.

The file cabinet contains four uniform drawers, each 2.5 ft long and weighing 30 lb/ft. If the empty cabinet (without drawers) has a weight of 40 lb and a center of mass at G, determine how many of...

Determine the force developed in the short link BD, and the tension in the cords CE and CF, and the reactions of the ball-and-socket joint A on the block. 3 m D 6 kN E 9 kN 1.5 m 4m

The truss is supported by a pin at A and a roller at B. Determine the support reactions. 5 kN 4 m 45 2 m 10 kN -2 m- -4 m-

Determine the reaction at fixed support A. A 6 m 0.5 kN/m -3 m- 3 kN 5 kN-m

The 2.5-Mg assembly has a center of mass at G and is hoisted with a constant velocity. Determine the force in each cable and the distance d for equilibrium. 3 m 1 m/ - 4 m P .G 4 m C B 4 m 1.25 m

Determine the magnitude of the moment of the force F = {50i - 20j - 80k} N about the baseline BC of the tripod. X 1.5 m 2 m 2m C 0.5 m B D F N 1 m 4 m A 2.5 m y

The curved pipe has a radius of 5 ft. If a force of 80 lb acts at its end as shown, determine the moment of this force about point C. Solve the problem by using two different position vectors. X 75-...

Using a ring collar the 75-N force can act in the vertical plane at various angles . Determine the magnitude of the moment it produces about point A, plt the result of M (ordinate) versus (abscissa)...

Replace the loading system acting on the post by an equivalent resultant force and couple moment at point O. 60 lb 3 4 P O 30 lb 1 ft 3 ft 2 ft 40 lb X

Determine the magnitude and orientation of of force F and its placement d on the beam so the loading system is equivalent to a resultant force of 20 kN acting vertically downward at point O and a...

Replace the force and couple moment system acting on the beam by an equivalent resultant force and couple moment at point O. 0.2 m -1.5m 30 500 N -2m 15 -1.5 m- 200 N 200 N-m X

Replace the forces acting on the gear by an equivalent resultant force and couple moment acting at point O. 2.25 kN 175 mm 41 60 3 kN 20 175 mm w X

Determine the horizontal and vertical components of force acting at the pins A, B, and C of the frame. 4 ft 45 4 ft 100 lb 4 ft 100 lb 8 ft 30%

Determine the horizontal and vertical components of force that pins A and B exert on the frame. 4 m A -3 m- B C 2 kN/m

Determine the force in each member of the truss. State if the members are in tension or compression. 600 16 450 lb YE A -3 ft- D B 30 -3 ft

Determine the force in each member of the truss and state if the members are in tension or compression. Set P 1 = 2 kN and P 2 = 1.5 kN. A 30 - 3 m B D P 30 3 m C P

Express the moment of the couple acting on the pipe assembly in Cartesian vector form. What is the magnitude of the couple moment? X Z 3 ft 1.5 ft X 1 ft. 20 lb. B 20 lb 1 ft 2 ft

If F = 80N, determine the magnitude and coordinate direction angles of the couple moment. The pipe assembly lies in the xy plane. X F 3 in. 3 in. 4 in. 10 lb Z -F 10 lb 30 -P 6 in. y

Three couple moments act on the pipe assembly. Determine the magnitudes of M 2 and M 3 so that the resultant couple moment is zero. My 0-2 120 120 120* M M-80N m

The hanger H is used to support the end of the floor beam B. If the loading on the hanger consists of two couples, determine the magnitude of the horizontal forces F and - F so that the resultant...

The cable exerts a 140-N force on the telephone pole. Determine the moment of this force about point A. Solve the problem using two different position vectors. F = 140 N/ 3 m B Z 6 m 2m -y

The force F = {6i + 8j + 10k} creates a moment about point O of M O = {-14i + 8j + 2k} N m. If the force passes through a point having an x coordinate of 1 m, determine the y and z coordinates of...

A force of F = {6i - 2j + 1k} k N produces a moment M O = {4i + 5j - 14k} of about the origin of coordinates, point O. If the force acts at a point having an x coordinate of x = 1 m, determine the y...

The pipe assembly is subjected to the 80-N force. Determine the moment of this force about point B. + 200 mm 30 N F = 80 N 400 mm 40 250 mm 300 mm 200 mm y

Determine the moment of the force of F = 600 N about point A. X 6 m B 45 4 m F N A 4 m 6 m

Determine the moment of the force F about point P. Express the result as a Cartesian vector. P X 2 m -3 m- 0 3 m- T A 1 m F = {2i + 4j6k) kN 3 m

The man pulls on the rope with a force of F = 20 N. Determine the moment of this force about the base of the pole at O. Solve the problem two ways, i.e., by using a position vector from O to A, then...

Determine the moment of the force F about point P. Express the result as a Cartesian vector. F = (-6i+4j+ 8k) kN 6 m x A -2 m- O N 4 m 1m P 3 m

Determine the smallest force F that must be applied to the rope in order to create a moment of M = 900 N m at point O. x B 1.5 m m F A 10.5 m

Determine the couple moment acting on the pipe assembly and express the result as a Cartesian vector. x B FA = 450 N 0.4 m 35 4 5 0.3 m 0 FB = 450 N C

The force F = {400i - 100j - 700} Ib acts at the end of the beam. Determine the moment of this force about point A. T 1.5 ft B -0.25 ft 8 ft N

A force of 80 N acts on the handle of the paper cutter at A. Determine the moment created by this force about the hinge at O, if = 60 At what angle should the force be applied so that the moment it...

If the man at B exerts a force of p = 30 Ib on his rope, determine the magnitude of the force F the man at C must exert to prevent the pole from rotating, i.e., so the resultant moment about A of...

Determine the moment of the force F about point O. Express the result as a Cartesian vector. F = (-6i+4j+ 8k) kN 6 m X -2 m- N 4 m 1m P 3 m

The cable exerts a force of P = 6 kN at the end of the 8-m-long crane boom. If = 30, determine the placement x of the hook at B so that this force creates a maximum moment about point O. What is...

If F B = 30 and F c = 45 Ib determine the resultant moment about the bolt located at A. 2.5 ft 20 FB 0.75 ft 30 Fc

The man exerts the wo forces on the handle of the shovel. Determine the resultant moment of these forces about the blade at A. 800 mm 65 450 mm F = 12 N F = 30 N 30 20

Region 1 is defined by x y + 2z > 5 with 1 = 2 o , while region 2 is defined by x y + 2z < 5 with 2 = 5 o . If H 1 = 40a x + 20a y 30a z A/m, find (a) H 1n , (b) H 2t ,

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 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...

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

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...

Show that a current density with vector potential A(r) = f (r)r has zero magnetic dipole moment.