Given the three nonzero vectors A, B, and C, show that if ,A (B X C) = 0 the three vectors must lie in the same plane.

Determine the coordinate direction angles of the force F applied at the end of the pipe such that the moment of F about point A is zero. 1 ft -A -2 ft- 3 ft 45 B F

If F = {550i + 60j + 30k} lb, determine the magnitude and coordinate direction angles of the moment of F about point A. 1 ft, z A -2 ft- 3 ft 45 B -y

The force F = {400i - 100j - 700k} lb acts at the end of the beam. Determine the moment of this force about point O. 1.5 ft X B -0.25 ft F 8 ft

In order to pull out the nail at B, the force F exerted on the handle of the hammer must produce a clockwise moment of 500 lbin. about point A. Determine the required magnitude of force F. 18 in. 30...

The tower crane is used to hoist a 2-Mg load upward at constant velocity. The 1.5-Mg jib BD and 0.5-Mg jib BC have centers of mass at G 1 and G 2 , respectively. Determine the required mass of the...

If the tension in the belt is 52 lb, determine the moment of each of these forces about the pin at A. 8 in. A 30 20 -5 in.- F = 52 lb 6 in. MAFU F = 52 lb

Determine the orientation (0 < < 180) of the 40-lb force F so that it produces (a) The maximum moment about point A (b) No moment about point A. Find the moment in case (a). 8 ft- 2 ft

The cable exerts a force of P = 6 kN at the end of the 8-m-long crane boom. If x = 10 m, determine the position of the boom so that this force creates a maximum moment about point O. What is this...

Two forces act on the skew caster. Determine the resultant moment of these forces about point A and about point B. 1.25 in. 2.5 in. 60 80 lb B 10 lb 2 in.

Determine the angle at which the 500-N force must act at A so that the moment of this force about point B is equal to zero. Im B 2 m- 10.3 E 500 N

Determine the shear force and moment acting at point D of the beam. - 6 ft- 9 ft D 2 kip/ft 9 ft- B

The beam-column is fixed to the floor and supports the load shown. Determine the normal force, shear force, and moment at points A and B. 5 kN 0.6 m 0.2 m 2m A B 0.8 kN. m 2.5 kN 0.4 m

The pliers are used to grip the tube at B. If a force of 20 lb is applied to the handles, determine the shear force and moment at point C. Assume the jaws of the pliers exert only normal forces on...

The structural connections transmit the loads shown to the column. Determine the normal force, shear force, and moment acting in the column at a section passing horizontally through point A. 30 mm 16...

Determine the normal force, shear force, and moment in the shaft at points B and D. These points lie just to the right of the 150-lb force and the bearing at C, respectively. There is a thrust...

Determine the moments of inertia for the shaded area with respect to the u and v axes. v y 0.5 in. 5 in. 1 in. 30 4 in.- u -0.5 in. 10.5 in. X

Determine the moment of inertia for the beams crosssectional area about the x axis that passes through the centroid C of the cross section. N 0] 25 mm 200 mm, 45 45 25 mm 100 mm 100 mm C 200 mm 45% 45

Determine which locates the centroid C of the crosssectional area of the wing channel, and then determine the moment of inertia I x about the centroidal x axis. Neglect the effect of rounded...

Determine the moment of inertia of the beams crosssectional area about the y axis. 30 mm 140 mm 30 mm 30 mm K 70 mm 30 mm 170 mm -x X

Determine the moment of inertia of the beams crosssectional area about the x axis. 30 mm 140 mm 30 mm y Js 30 mm 70 mm 30 mm 170 mm X

Determine the moment of inertia of the beams crosssectional area about the x axis. 200 mm y X. 14 100 mm C 300 mm 50 mm 50 mm -X

Determine the moment of inertia of the area about the x axis. T 3 in. 6 in. y 6 in. 6 in.- X

Determine the moment of inertia of the beams cross-sectional area about the x-axis. 6 in. y 2 in. y' 4 in.. C 1 in. 1 in. 1 in. 1 in. X

Determine the moment of inertia of the equilateral triangle about the x axis passing through its centroid. y C a - y = 3 (z - x) a

Determine the moment of inertia for the area about the y axis. y y = = h 6343 -b h X

Determine the magnitude of the force P and the orientation of the 200-lb force required to keep the particle in equilibrium. (-1 ft, -7 ft, 4 ft) F3= 200 lb X N F=360 lb F= 120 lb 20 F = 300 lb P y

Determine the friction developed between the 50-kg crate and the ground if (a) P = 200 N, and (b) P = 400 N. The coefficients of static and kinetic friction between the crate and the ground are s =...

Determine the components of reaction acting at the balland- socket A, roller B, and cord CD. X B 400 N 2 m 2 m Z D 300 N 1m 2 m C y

Locate the centroid x of the area. y y = a h a-x +h h X

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

Locate the centroid (x, y) of the exparabolic segment of area. b 4 y X

Determine the magnitude of the hydrostatic force acting on gate AB, which has a width of 4 ft. The specific weight of water is = 62.4 lb/ft. -3 ft- 4 ft

Locate the centroid of the area. a y -L- -y = a sin x L X

Locate the centroid x of the area. y 4 m y = 4 - 1x -8 m- X

Locate the centroid y of the area. y 4 m 1 x = 4 = 1/6 x -8 m-

Locate the centroid y of the area. y y = -b- h -X

Locate the centroid y of the area. 4 m 4 m -X

Locate the centroid x of the area. 4m (y=4x -4m

Locate the centroid y of the area. 1 1 m y = 1-1/x (Y=1 -2 m- X

Locate the centroid (x, y, z) of the wire bent in the shape shown. 300 mm 600 mm z 400 mm

Locate the centroid x of the shaded area. y a q xn h X

Locate the centroid y of the homogeneous solid formed by revolving the shaded area about the y axis. -1 m- -2= y 0.5 m -y

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

Determine the centroid (x,y) of the area. 1m y = r 1 m X

Determine the centroid (x,y) of the area. 1m -y=x -1m- -X

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

A cone is attached to the hemisphere. If both pieces have the same density, determine the height h of the cone if the configuration is to be in neutral equilibrium. 3 ft h

The door has a uniform weight of 50 lb. It is hinged at A and is held open by the 30-lb weight and the pulley. Determine the angle for equilibrium. A k = 200 N/m wwwww 0 40 Nm 0.5 m

Consider a particle in the potential shown in Figure 7.3. (a) Find the first-order correction to the ground-state wave function. The first three nonzero terms in the sum will suffice. (b) Using the...

Determine the magnitude of force P required to hold the 50-kg smooth rod in equilibrium at = 60. B 5 m 0 Prob. F11-2 A P

Determine the moment of inertia of the area about the y axis. y = 2 cos(x) -4 in.- y -4 in.- 2 in. X

Determine the moment of inertia for the area about the x axis y y = 4x- - 4 in. 4 in. --x

Determine the moment of inertia of the area about the y axis. 80 mm y 20 mm -y =(400-x) X

Locate the centroid of the area. Solve the problem by evaluating the integrals using Simpsons rule. y y = 0.5ex 1 m -X

The load can be supported by two boys when the cord is suspended over the pipe (a half-turn). If each boy can pull with a force of 125 lb, determine the maximum weight of the load. Can one boy...

The hand clamp is constructed using a square-threaded screw having a mean diameter of 36 mm, a lead of 4 mm, and a coefficient of static friction at the screw of s = 0.3. To tighten the screw, a...

Determine the moment of inertia I x of the area about the x axis. y 150 mm 150 mm O -100 mm-100 mm-150 mm - 75 mm X

Draw the free-body diagram for the following problems. a) The rod in Prob. 525. b) The bar in Prob. 527. c) The disk in Prob. 528. 3 -3 ft- 4 100 lb + -3 ft- B 200 lb-ft 2 ft 13 12

Draw the free-body diagram for the following problems. a) The clamp in Prob. 532. b) The jib crane in Prob. 533. c) The crane in Prob. 535. d) The beam in Prob. 536. -150 mm- B 250 mm

(a) The potential takes the constant value 0 on the closed surface S which bounds the volume V. (b) The total charge inside V is Q. There is no charge anywhere else. Show that the electrostatic...

(a) Evaluate the relevant part of the integral to find the interaction energy V E between two identical insulating spheres, each with radius R and charge Q distributed uniformly over their surfaces....

Let the space between two concentric spheres with radii a and R a be filled uniformly with charge. (a) Calculate the total energy U E in terms of the total charge Q and the variable x = a/R. Check...

The figure below shows a cube filled uniformly with charge. Determine the ratio 0 / 1 of the potential at the center of the cube to the potential at the corner of the cube. -% S

Let d and s be two unequal lengths. Assume that charge is distributed on the z = 0 plane with a surface density (a) Integrate to find the total charge Q on the plane. (b) Show that the potential (z)...

Suppose that the electrostatic potential produced by a point charge was not Coulombic, but instead varied with distance in a manner determined by a specified scalar function f (r): (a) Calculate the...

(a) Show that the electric field produced by a uniform charge density confined to the volume V enclosed by a surface S can be written (b) Show that the electric field due to an arbitrary but...

Suppose the electrostatic potential of a point charge were (r) = (1/4 0 )r (1+) rather than the usual Coulomb formula. (a) Find the potential (r) at a point at a distance r from the center of a...

The figure below shows a circular hole of radius b (white) bored through a spherical shell (gray) with radius R and uniform charge per unit area . (a) Show that E(P) = (/2 0 )[1 sin( 0 /2)]r, where...

A two-dimensional disk of radius R carries a uniform charge per unit area > 0. (a) Calculate the potential at any point on the symmetry axis of the disk. (b) Calculate the potential at any point on...

(a) Use Greens second identity, V d 3 r (f 2 g g 2 f ) = S dS (f g g f), to prove that the potential (0) at the center of a charge-free spherical volume V is equal to the average of (r) over the...

A charge distribution (r) with total charge Q occupies a finite volume V somewhere in the half-space z < 0. If the integration surface is z = 0, prove that z=0 dS 2. E = ala . 20

(a) Find E(r) if (x, y, z) = 0 (x) + 0(x) 0(x b). (b) Show by explicit calculation that (x, y, z) does not exert a net force on itself.

The z-axis coincides with the symmetry axis of a flat disk of radius a in the x-y plane. The disk carries a uniform charge per unit area < 0. The rim of the disk carries an additional uniform charge...

Show that the torque exerted on a charge distribution (r) by a distinct charge distribution ' (r') is N = 1 471060 [ar [d farfar rxr r'/0 (r)p'(r').

Use Gauss law to find the electric field when the charge density is: (a) Expresses the answer in Cartesian coordinates. (b) Express the answer in cylindrical coordinates. (c) Express the answer in...

Draw the electric field line pattern for a line of five equally spaced charges with equal magnitude but alternating algebraic signs, as sketched below. Be sure to choose the scale of your drawing and...

The Cartesian components of the electric field in a charge-free region of space are E k = C k + D jk r j , where C k and D jk are constants. (a) Prove that D jk is symmetric (D jk = D kj ) and...

(a) A point charge q > 0 with total energy E travels through a region of constant potential V1 and enters a region of potential V2 < V1. Show that the trajectory bends so that the angles 1 and 2 in...

A point particle with charge q and mass m is fixed at the origin. An identical particle is released from rest at x = d. Find the asymptotic (x ) speed of the released particle.

If is a real constant, the continuity equation is satisfied by the charge and current distributions. The given j represents current flowing in toward the origin of coordinates. But the given is...

An infinitely long cylindrical solenoid carries a spatially uniform but time-dependent surface current density K(t) = K 0 (t/). K 0 and are constants. Find the electric and magnetic fields...

Let be a parameter and define new electric and magnetic field vectors as linear combinations of the usual electric and magnetic field vectors: E' = E' cos + cBsin cB' = E sin + cB cos . (a) Show...

If the photon had a mass m, Gauss law with E = changes from 2 = / 0 to an equation which includes a lenght L = /mc: Experimental searches for m use a geometry first employed by Cavendish where a...

In 1942, Boris Podolsky proposed a generalization of electrostatics that eliminates the divergence of the Coulomb field for a point charge. His theory retains E = 0 but replaces Gauss law by (a)...

A surface current density K(r S , t) flows in the z = 0 plane which separates region 1 (z > 0) from region 2 (z < 0). Each region contains arbitrary, time-dependent distributions of charge and...

Consider a collection of point particles fixed in space with charge density is : Suppose that : E(r, t = 0) = B(r, t = 0) = 0 and (a) Construct a simple current density which satisfies the continuity...

A particle with charge q is confined to the x-y plane and sits at rest somewhere away from the origin until t = 0. At that moment, a magnetic field B(x, y) = (x)(y) turns on with a value of which...

The magnetostatic equation B = 0 j is not consistent with conservation of charge for a general time-dependent charge density. Show that consistency can be achieved using B = 0 j + j D and a...

The electric and magnetic fields for time-independent distributions of charge and current which go to zero at infinity are (a) Calculate E and E. (b) Calculate B and B. The curl calculation...

Let F 1 and F 2 be the instantaneous forces that act on a particle with charge q when it moves through a magnetic field B(r) with velocities 1 and 2 , respectively. Without choosing a coordinate...

Let r 1 (r 2 ) point to a line element ds 1 (ds 2 ) of a closed loop C 1 (C 2 ) which carries a current I 1 (I 2 ). Experiment shows that the force exerted on I 1 by I 2 is (a) Show that (b) Use (a)...

Let S be the surface that bounds a volume V . Show that (a) s dS = 0; (b) 1/3 s dS r = V.

Compute the unit normal vector n to the ellipsoidal surfaces defined by constant values of Check that you get the expected answer when a = b = c. (x, y, z) = V + 6 2

The buckets on the conveyor travel with a speed of 15 ft/s. Each bucket contains a block which falls out of the bucket when = 120. Determine the distance s to where the block strikes the conveyor....

Particles A and B are traveling around a circular track at a speed of 8 m/s at the instant shown. If the speed of B is increasing by (a t ) B = 4 m/s 2 , and at the same instant A has an increase in...

The boat is originally traveling at a speed of 8 m/s when it is subjected to the acceleration shown in the graph. Determine the boats maximum speed and the time t when it stops. - t(s) 1 24 9 + 1//3...

Point charges q 1 , q 2 , . . ., q N are embedded in a body with permittivity in . The latter is itself embedded in a body with permittivity out . Find the total polarization charge Q pol induced...

Write the Helmholtz theorem expression for D(r) and eliminate D itself from the integrals you write down. How does this formula simplify (if at all) for simple dielectric matter?

A metal ball with charge Q sits at the center of a thin, spherical, conducting shell. The shell has charge Q' and the space between the shell and the ball is filled with matter with dielectric...