Solved: A homogeneous wire is bent into the shape shown. Determine by direct integration

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

Determine the x coordinate of the centroid of the trapezoid shown in terms of h1, h2, and a.

For the plane area of Problem 5.5, determine the ratio a/r so that the centroid of the area is located at point B.

Determine the y coordinate of the centroid of the shaded area in terms of r1, r2, and .

Show that as r1 approaches r2, the location of the centroid approaches that for an arc of circle of radius 12 ( )/2.

The horizontal x-axis is drawn through the centroid C of the area shown, and it divides the area into two component areas A1 and A2. Determine the first moment of each component area with respect to the x-axis, and explain the results obtained.

The horizontal x-axis is drawn through the centroid C of the area shown, and it divides the area into two component areas A1 and A2. Determine the first moment of each component area with respect to the x-axis, and explain the results obtained.

A composite beam is constructed by bolting four plates to four 60 60 12-mm angles as shown. The bolts are equally spaced along the beam, and the beam supports a vertical load. As proved in mechanics of materials, the shearing forces exerted on the bolts at A and B are proportional to the first moments with respect to the centroidal x-axis of the red-shaded areas shown, respectively, in parts a and b of the figure. Knowing that the force exerted on the bolt at A is 280 N, determine the force exerted on the bolt at B.

The first moment of the shaded area with respect to the x-axis is denoted by Qx. (a) Express Qx in terms of b, c, and the distance y from the base of the shaded area to the x-axis. (b) For what value of y is x Q maximum, and what is that maximum value?

A thin, homogeneous wire is bent to form the perimeter of the figure indicated. Locate the center of gravity of the wire figure thus formed.

A thin, homogeneous wire is bent to form the perimeter of the figure indicated. Locate the center of gravity of the wire figure thus formed.

A thin, homogeneous wire is bent to form the perimeter of the figure indicated. Locate the center of gravity of the wire figure thus formed.

A thin, homogeneous wire is bent to form the perimeter of the figure indicated. Locate the center of gravity of the wire figure thus formed.

The homogeneous wire ABCD is bent as shown and is attached to a hinge at C. Determine the length L for which portion BCD of the wire is horizontal.

The homogeneous wire ABCD is bent as shown and is attached to a hinge at C. Determine the length L for which portion AB of the wire is horizontal.

The homogeneous wire ABC is bent into a semicircular arc and a straight section as shown and is attached to a hinge at A. Determine the value of for which the wire is in equilibrium for the indicated position.

A uniform circular rod of weight 8 lb and radius 10 in. is attached to a pin at C and to the cable AB. Determine (a) the tension in the cable, (b) the reaction at C.

Determine the distance h for which the centroid of the shaded area is as far above line BB as possible when (a) k = 0.10, (b) k = 0.80.

Knowing that the distance h has been selected to maximize the distance y from line BB to the centroid of the shaded area, show that 2 /3. yh

Determine by direct integration the centroid of the area shown. Express your answer in terms of a and h.

Determine by direct integration the centroid of the area shown. Express your answer in terms of a and h.

Determine by direct integration the centroid of the area shown. Express your answer in terms of a and h.

Determine by direct integration the centroid of the area shown.

Determine by direct integration the centroid of the area shown.

Determine by direct integration the centroid of the area shown.

Determine by direct integration the centroid of the area shown. Express your answer in terms of a and b.

Determine by direct integration the centroid of the area shown. Express your answer in terms of a and b.

Determine by direct integration the centroid of the area shown.

Determine by direct integration the centroid of the area shown. Express your answer in terms of a and b.

Determine by direct integration the centroid of the area shown. Express your answer in terms of a and b.

A homogeneous wire is bent into the shape shown. Determine by direct integration the x coordinate of its centroid.

A homogeneous wire is bent into the shape shown. Determine by direct integration the x coordinate of its centroid.

A homogeneous wire is bent into the shape shown. Determine by direct integration the x coordinate of its centroid. Express your answer in terms of a.

Determine by direct integration the centroid of the area shown.

Determine by direct integration the centroid of the area shown.

Determine the centroid of the area shown when 2 a = in.

Determine the value of a for which the ratio /xy is 9.

Determine the volume and the surface area of the solid obtained by rotating the area of Problem 5.1 about (a) the x-axis, (b) the y-axis.

Determine the volume and the surface area of the solid obtained by rotating the area of Problem 5.2 about (a) the line y = 72 mm, (b) the x-axis.

Determine the volume and the surface area of the solid obtained by rotating the area of Problem 5.8 about (a) the line x = 60 mm, (b) the line y = 120 mm.

Determine the volume of the solid generated by rotating the parabolic area shown about (a) the x-axis, (b) the axis AA.

Determine the volume and the surface area of the chain link shown, which is made from a 6-mm-diameter bar, if R = 10 mm and L = 30 mm.

Verify that the expressions for the volumes of the first four shapes in Figure 5.21 on Page 264 are correct.

Determine the volume and weight of the solid brass knob shown, knowing that the specific weight of brass is 0.306 lb/in3.

Determine the total surface area of the solid brass knob shown.

The aluminum shade for the small high-intensity lamp shown has a uniform thickness of 1 mm. Knowing that the density of aluminum is 2800 kg/m3, determine the mass of the shade.

The escutcheon (a decorative plate placed on a pipe where the pipe exits from a wall) shown is cast from brass. Knowing that the density of brass is 8470 kg/m3, determine the mass of the escutcheon.

A 3 4 -in.-diameter hole is drilled in a piece of 1-in.-thick steel; the hole is then countersunk as shown. Determine the volume of steel removed during the countersinking process.

Knowing that two equal caps have been removed from a 10-in.-diameter wooden sphere, determine the total surface area of the remaining portion.

Determine the capacity, in liters, of the punch bowl shown if R = 250 mm.

The shade for a wall-mounted light is formed from a thin sheet of translucent plastic. Determine the surface area of the outside of the shade, knowing that it has the parabolic cross section shown.

For the beam and loading shown, determine (a) the magnitude and location of the resultant of the distributed load, (b) the reactions at the beam supports.

For the beam and loading shown, determine (a) the magnitude and location of the resultant of the distributed load, (b) the reactions at the beam supports.

Determine the reactions at the beam supports for the given loading.

Determine the reactions at the beam supports for the given loading.

Determine the reactions at the beam supports for the given loading.

Determine the reactions at the beam supports for the given loading.

Determine the reactions at the beam supports for the given loading.

Determine the reactions at the beam supports for the given loading.

Determine the reactions at the beam supports for the given loading when wO = 400 lb/ft.

Determine (a) the distributed load wO at the end A of the beam ABC for which the reaction at C is zero, (b) the corresponding reaction at B.

Determine (a) the distance a so that the vertical reactions at supports A and B are equal, (b) the corresponding reactions at the supports.

Determine (a) the distance a so that the reaction at support B is minimum, (b) the corresponding reactions at the supports.

A beam is subjected to a linearly distributed downward load and rests on two wide supports BC and DE, which exert uniformly distributed upward loads as shown. Determine the values of wBC and wDE corresponding to equilibrium when 600 Aw = N/m.

A beam is subjected to a linearly distributed downward load and rests on two wide supports BC and DE, which exert uniformly distributed upward loads as shown. Determine (a) the value of wA so that wBC = wDE, (b) the corresponding values of wBC and wDE.

The cross section of a concrete dam is as shown. For a 1-m-wide dam section, determine (a) the resultant of the reaction forces exerted by the ground on the base AB of the dam, (b) the point of application of the resultant of part a, (c) the resultant of the pressure forces exerted by the water on the face BC of the dam.

The cross section of a concrete dam is as shown. For a 1-m-wide dam section, determine (a) the resultant of the reaction forces exerted by the ground on the base AB of the dam, (b) the point of application of the resultant of part a, (c) the resultant of the pressure forces exerted by the water on the face BC of the dam.

An automatic valve consists of a 9 9-in. square plate that is pivoted about a horizontal axis through A located at a distance h = 3.6 in. above the lower edge. Determine the depth of water d for which the valve will open.

An automatic valve consists of a 9 9-in. square plate that is pivoted about a horizontal axis through A. If the valve is to open when the depth of water is d = 18 in., determine the distance h from the bottom of the valve to the pivot A.

The 3 4-m side AB of a tank is hinged at its bottom A and is held in place by a thin rod BC. The maximum tensile force the rod can withstand without breaking is 200 kN, and the design specifications require the force in the rod not to exceed 20 percent of this value. If the tank is slowly filled with water, determine the maximum allowable depth of water d in the tank.

The 3 4-m side of an open tank is hinged at its bottom A and is held in place by a thin rod BC. The tank is to be filled with glycerine, whose density is 1263 kg/m3. Determine the force T in the rod and the reactions at the hinge after the tank is filled to a depth of 2.9 m.

The friction force between a 6 6-ft square sluice gate AB and its guides is equal to 10 percent of the resultant of the pressure forces exerted by the water on the face of the gate. Determine the initial force needed to lift the gate if it weighs 1000 lb.

A tank is divided into two sections by a 1 1-m square gate that is hinged at A. A couple of magnitude 490 N m is required for the gate to rotate. If one side of the tank is filled with water at the rate of 0.1 m3/min and the other side is filled simultaneously with methyl alcohol (density ma = 789 kg/m3) at the rate of 0.2 m3/min, determine at what time and in which direction the gate will rotate.

A prismatically shaped gate placed at the end of a freshwater channel is supported by a pin and bracket at A and rests on a frictionless support at B. The pin is located at a distance 0.10 m h = below the center of gravity C of the gate. Determine the depth of water d for which the gate will open.

A prismatically shaped gate placed at the end of a freshwater channel is supported by a pin and bracket at A and rests on a frictionless support at B. Determine the distance h if the gate is to open when 0.75 m. d =

The square gate AB is held in the position shown by hinges along its top edge A and by a shear pin at B. For a depth of water 3.5 d = ft, determine the force exerted on the gate by the shear pin.

A long trough is supported by a continuous hinge along its lower edge and by a series of horizontal cables attached to its upper edge. Determine the tension in each of the cables, at a time when the trough is completely full of water.

A 0.5 0.8-m gate AB is located at the bottom of a tank filled with water. The gate is hinged along its top edge A and rests on a frictionless stop at B. Determine the reactions at A and B when cable BCD is slack.

A 0.5 0.8-m gate AB is located at the bottom of a tank filled with water. The gate is hinged along its top edge A and rests on a frictionless stop at B. Determine the minimum tension required in cable BCD to open the gate.

A 4 2-ft gate is hinged at A and is held in position by rod CD. End D rests against a spring whose constant is 828 lb/ft. The spring is undeformed when the gate is vertical. Assuming that the force exerted by rod CD on the gate remains horizontal, determine the minimum depth of water d for which the bottom B of the gate will move to the end of the cylindrical portion of the floor.

Solve Problem 5.94 if the gate weighs 1000 lb. PROBLEM 5.94 A 4 2-ft gate is hinged at A and is held in position by rod CD. End D rests against a spring whose constant is 828 lb/ft. The spring is undeformed when the gate is vertical. Assuming that the force exerted by rod CD on the gate remains horizontal, determine the minimum depth of water d for which the bottom B of the gate will move to the end of the cylindrical portion of the floor.

A hemisphere and a cone are attached as shown. Determine the location of the centroid of the composite body when (a) 1.5 , = ha (b) 2. = ha

Consider the composite body shown. Determine (a) the value of x when /2, hL = (b) the ratio h/L for which . xL

Determine the y coordinate of the centroid of the body shown.

Determine the z coordinate of the centroid of the body shown. (Hint: Use the result of Sample Problem 5.13.)

For the machine element shown, locate the y coordinate of the center of gravity.

For the machine element shown, locate the y coordinate of the center of gravity.

For the machine element shown, locate the x coordinate of the center of gravity.

For the machine element shown, locate the z coordinate of the center of gravity.

For the machine element shown, locate the x coordinate of the center of gravity.

For the machine element shown, locate the z coordinate of the center of gravity.

Locate the center of gravity of the sheet-metal form shown.

Locate the center of gravity of the sheet-metal form shown.

A window awning is fabricated from sheet metal of uniform thickness. Locate the center of gravity of the awning.

A thin sheet of plastic of uniform thickness is bent to form a desk organizer. Locate the center of gravity of the organizer.

A wastebasket, designed to fit in the corner of a room, is 16 in. high and has a base in the shape of a quarter circle of radius 10 in. Locate the center of gravity of the wastebasket, knowing that it is made of sheet metal of uniform thickness.

A mounting bracket for electronic components is formed from sheet metal of uniform thickness. Locate the center of gravity of the bracket.

An 8-in.-diameter cylindrical duct and a 4 8-in. rectangular duct are to be joined as indicated. Knowing that the ducts were fabricated from the same sheet metal, which is of uniform thickness, locate the center of gravity of the assembly.

An elbow for the duct of a ventilating system is made of sheet metal of uniform thickness. Locate the center of gravity of the elbow.

Locate the center of gravity of the figure shown, knowing that it is made of thin brass rods of uniform diameter.

Locate the center of gravity of the figure shown, knowing that it is made of thin brass rods of uniform diameter.

A thin steel wire of uniform cross section is bent into the shape shown. Locate its center of gravity.

The frame of a greenhouse is constructed from uniform aluminum channels. Locate the center of gravity of the portion of the frame shown.

Three brass plates are brazed to a steel pipe to form the flagpole base shown. Knowing that the pipe has a wall thickness of 8 mm and that each plate is 6 mm thick, determine the location of the center of gravity of the base. (Densities: brass = 8470 kg/m3; steel = 7860 kg/m3.)

A brass collar, of length 2.5 in., is mounted on an aluminum rod of length 4 in. Locate the center of gravity of the composite body. (Specific weights: brass = 0.306 lb/in3, aluminum = 0.101 lb/in3)

A bronze bushing is mounted inside a steel sleeve. Knowing that the specific weight of bronze is 0.318 lb/in3 and of steel is 0.284 lb/in3, determine the location of the center of gravity of the assembly.

A scratch awl has a plastic handle and a steel blade and shank. Knowing that the density of plastic is 1030 kg/m3 and of steel is 3 7860 kg/m , locate the center of gravity of the awl.

Determine by direct integration the values of x for the two volumes obtained by passing a vertical cutting plane through the given shape of Figure 5.21. The cutting plane is parallel to the base of the given shape and divides the shape into two volumes of equal height. A hemisphere

Determine by direct integration the values of x for the two volumes obtained by passing a vertical cutting plane through the given shape of Figure 5.21. The cutting plane is parallel to the base of the given shape and divides the shape into two volumes of equal height. A semiellipsoid of revolution

Determine by direct integration the values of x for the two volumes obtained by passing a vertical cutting plane through the given shape of Figure 5.21. The cutting plane is parallel to the base of the given shape and divides the shape into two volumes of equal height. A paraboloid of revolution

Locate the centroid of the volume obtained by rotating the shaded area about the x-axis.

Locate the centroid of the volume obtained by rotating the shaded area about the x-axis.

Locate the centroid of the volume obtained by rotating the shaded area about the line . xh

Locate the centroid of the volume generated by revolving the portion of the sine curve shown about the x-axis.

Locate the centroid of the volume generated by revolving the portion of the sine curve shown about the y-axis. (Hint: Use a thin cylindrical shell of radius r and thickness dr as the element of volume.)

Show that for a regular pyramid of height h and n sides ( 3, 4, ) n= the centroid of the volume of the pyramid is located at a distance h/4 above the base.

Determine by direct integration the location of the centroid of one-half of a thin, uniform hemispherical shell of radius R.

The sides and the base of a punch bowl are of uniform thickness t. If tR << and R = 250 mm, determine the location of the center of gravity of (a) the bowl, (b) the punch.

Locate the centroid of the section shown, which was cut from a thin circular pipe by two oblique planes.

Locate the centroid of the section shown, which was cut from an elliptical cylinder by an oblique plane.

After grading a lot, a builder places four stakes to designate the corners of the slab for a house. To provide a firm, level base for the slab, the builder places a minimum of 3 in. of gravel beneath the slab. Determine the volume of gravel needed and the x coordinate of the centroid of the volume of the gravel. (Hint: The bottom surface of the gravel is an oblique plane, which can be represented by the equation y = a + bx + cz.)

Determine by direct integration the location of the centroid of the volume between the xz plane and the portion shown of the surface y = 16h(ax x2)(bz z2)/a2b2.

Locate the centroid of the plane area shown.

Locate the centroid of the plane area shown.

The frame for a sign is fabricated from thin, flat steel bar stock of mass per unit length 4.73 kg/m. The frame is supported by a pin at C and by a cable AB. Determine (a) the tension in the cable, (b) the reaction at C.

Determine by direct integration the centroid of the area shown. Express your answer in terms of a and h.

Determine by direct integration the centroid of the area shown.

Three different drive belt profiles are to be studied. If at any given time each belt makes contact with one-half of the circumference of its pulley, determine the contact area between the belt and the pulley for each design.

Determine the reactions at the beam supports for the given loading.

The beam AB supports two concentrated loads and rests on soil that exerts a linearly distributed upward load as shown. Determine the values of A and B corresponding to equilibrium.

The base of a dam for a lake is designed to resist up to 120 percent of the horizontal force of the water. After construction, it is found that silt (that is equivalent to a liquid of density 33 1.76 10 kg/m )s = is settling on the lake bottom at the rate of 12 mm/year. Considering a 1-m-wide section of dam, determine the number of years until the dam becomes unsafe.

Determine the location of the centroid of the composite body shown when (a) 2, hb = (b) 2.5 . hb

Locate the center of gravity of the sheet-metal form shown.

Locate the centroid of the volume obtained by rotating the shaded area about the x-axis.