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

(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...

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

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...

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

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) 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...

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...

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...

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...

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...

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...

The text proved that the force on an isolated dielectric is Where E(r) is the total field at an interior point r and E avg (r S ) is the average of the total field just inside and just outside the...

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 dielectric body with permittivity in is embedded in an infinite volume of dielectric matter with permittivity out . The entire system is polarized by an external electric field E ext . If is the...

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...

The parallel-plate capacitor shown below is made of two identical conducting plates of area A carrying charges q. The capacitor is filled with a compressible dielectric solid with permittivity and...

An origin-centered sphere with permittivity and radius a is placed in a uniform external electric field E 0 . What radius b < a should an origin-centered, perfectly conducting sphere similarly...

A polarizable sphere of radius R is filled with free charge with uniform density c . The dielectric constant of the sphere is . (a) Find the polarization P(r). (b) Confirm explicitly that the total...

Students are often told that E = F q /q defines the electric field at a point if F q is the measured force on a tiny charge q placed at that point. More careful instructors let q 0 to avoid the...

An air-gap capacitor with parallel-plate area A discharges by the electrical breakdown of the air between its parallel plates (separation d) when the voltage between its plates exceeds V 0 . Lay a...

The electrostatic polarization inside an origin-centered sphere is P(r) = P(r). (a) Show that (r) outside the sphere is equal to the potential of a point electric dipole at the origin with a moment...

A cube is polarized uniformly parallel to one of its edges. Show that the electric field at the center of the cube is E(0) = P/3 0 . Compare with E(0) for a uniformly polarized sphere.

The polarization in all of space has the form P = P(r R)r, where P and R are constants. Find the polarization charge density and the electric field everywhere.

Find the total electrostatic energy of a ball with radius R and uniform polarization P.

Find a polarization P(r) which produces a polarization charge density in the form of an origin-centered sphere with radius R and uniform volume charge density P .

Two spheres with radius R have uniform but equal and opposite charge densities . The centers of the two spheres fail to coincide by an infinitesimal displacement vector . Show by direct superposition...

A conducting shell of radius R has total charge Q. If sawed in half, the two halves of the shell will fly apart. This can be prevented by placing a point charge Q' at the center of the shell. (a)...

(a) A spherical metal shell is charged to an electrostatic potential V . Cut this shell in half and pull the halves infinitesimally apart. Find the force with which one hemisphere of the shell repels...

A battery maintains the potential difference V between the spheres of a spherical capacitor with capacitance C. Move the center of the inner sphere away from the center of the outer sphere by an...

A long, straight wire has length L and a circular cross section with area a 2 . Arrange two such wires so they are parallel and separated by a distance d. You may assume that L >> d >> a and ignore...

Two pyramid-shaped conductors each carry a net charge Q. (a) Transfer charge Q from pyramid 2 to pyramid 1. Derive a condition on the coefficients of potential P ij which guarantees that this charge...

A non-conducting square has a fixed surface charge distribution. Make a rectangle with the same area and total charge by cutting off a slice from one side of the square and gluing it onto an adjacent...

A point electric dipole with moment p is placed at the center of a hollow spherical cavity scooped out of an infinite conducting medium. (a) Find the surface charge density induced on the surface of...

Let C be the capacitance of capacitor formed from two identical, flat conductor plates separated by a distance d. The plates have area A and arbitrary shape. When d < < A, we know that the...

A conducting disk of radius R held at potential V sits in the x-y plane centered on the z-axis. (a) Use the charge density for this system calculated in the text to find the potential everywhere on...