Solved: The tank is used to store a liquid having a

Locate the center of mass of the homogeneous rod bent into the shape of a circular arc.

Determine the location \((\bar{x}, \bar{y})\) of the centroid of the wire.

Locate the center of gravity \((\bar{x}\) of the homogeneous rod. If the rod has a weight per unit length of 100 N/m, determine the vertical reaction at A and the x and y components of reaction at the pin B.

Determine the distance \((\bar{y}\) to the center of gravity of the homogeneous rod

Locate the centroid \((\bar{y}\) of the area.

Locate the centroid \((\bar{x}\) of the parabolic area

Locate the centroid of the shaded area.

Locate the centroid \((\bar{x}\) of the shaded area

Locate the centroid \((\bar{y}\) of the shaded area.

Locate the centroid \((\bar{x}\) of the shaded area

Locate the centroid \((\bar{y}\) of the shaded area.

Locate the centroid \((\bar{x}\) of the shaded area

Locate the centroid \((\bar{y}\) of the shaded area.

Locate the centroid \((\bar{x}\) of the shaded area. Solve the problem by evaluating the integrals using Simpson’s rule.

Locate the centroid \((\bar{y}\) of the shaded area. Solve the problem by evaluating the integrals using Simpson’s rule.

Locate the centroid \((\bar{y}\) of the area

Locate the centroid \((\bar{x}\) of the area.

Locate the centroid \((\bar{y}\) of the area.

Locate the centroid \((\bar{y}\) of the shaded area

Locate the centroid \((\bar{x}\) of the shaded area.

Locate the centroid \((\bar{y}\) of the shaded area.

Locate the centroid \((\bar{x}\) of the shaded area.

Locate the centroid \((\bar{y}\) of the shaded area.

The plate has a thickness of 0.25 ft and a specific weight of \(\gamma\) = 180 lb/\(ft^{3}\). Determine the location of its center of gravity. Also, find the tension in each of the cords used to support it.

Locate the centroid \((\bar{x}\) of the shaded area.

Locate the centroid \((\bar{y}\) of the shaded area.

Locate the centroid \((\bar{x}\) of the shaded area.

Locate the centroid \((\bar{y}\) of the shaded area.

Locate the centroid \((\bar{x}\) of the shaded area.

Locate the centroid \((\bar{y}\) of the shaded area.

Locate the centroid \((\bar{x}\) of the shaded area.

Locate the centroid \((\bar{y}\) of the shaded area.

The steel plate is 0.3 m thick and has a density of 7850 kg/\(m^{3}\) . Determine the location of its center of mass. Also find the reactions at the pin and roller support.

Locate the centroid \((\bar{x}\) of the shaded area.

Locate the centroid \((\bar{y}\) of the shaded area.

Locate the centroid \((\bar{x}\) of the circular sector.

Determine the location \((\bar{r}\) of the centroid C for the loop of the lemniscate, \(r^{2}=2 a^{2} \cos 2 \theta,\left(-45^{\circ} \leq \theta \leq 45^{\circ}\right)\).

Locate the center of gravity of the volume. The material is homogeneous.

Locate the centroid \((\bar{y}\) of the paraboloid.

Locate the centroid \((\bar{z}\) of the frustum of the right-circular cone.

Determine the centroid \((\bar{y}\) of the solid.

Locate the centroid of the quarter-cone

The hemisphere of radius r is made from a stack of very thin plates such that the density varies with height, \(\rho\) = kz, where k is a constant. Determine its mass and the distance \((\bar{z}\) to the center of mass G.

Locate the centroid \((\bar{z}\) of the volume.

Locate the centroid of the ellipsoid of revolution.

Locate the center of gravity \((\bar{z}\) of the solid.

Locate the center of gravity \((\bar{y}\) of the volume. The material is homogeneous.

Locate the centroid \((\bar{z}\) of the spherical segment

Determine the location \((\bar{z}\) of the centroid for the tetrahedron. Suggestion: Use a triangular “plate” element parallel to the x–y plane and of thickness dz.

The truss is made from five members, each having a length of 4 m and a mass of 7 kg/m. If the mass of the gusset plates at the joints and the thickness of the members can be neglected, determine the distance d to where the hoisting cable must be attached, so that the truss does not tip (rotate) when it is lifted

Determine the location \((\bar{x}, \bar{y}, \bar{z})\) of the centroid of the homogeneous rod.

A rack is made from roll-formed sheet steel and has the cross section shown. Determine the location \((\bar{x}, \bar{y})\) of the centroid of the cross section. The dimensions are indicated at the center thickness of each segment.

Locate the centroid \((\bar{x}, \bar{y})\) of the metal cross section. Neglect the thickness of the material and slight bends at the corners

Locate the center of gravity \((\bar{x}, \bar{y}, \bar{z})\) of the homogeneous wire.

The steel and aluminum plate assembly is bolted together and fastened to the wall. Each plate has a constant width in the z direction of 200 mm and thickness of 20 mm. If the density of A and B is \(\rho_{s}=7.85 \mathrm{Mg} / \mathrm{m}^{3}\), and for C, \(\rho_{a l}=2.71 \mathrm{Mg} / \mathrm{m}^{3}\), determine the location \((\bar{x}\) of the center of mass. Neglect the size of the bolts

Locate the center of gravity G(x, y) of the streetlight. Neglect the thickness of each segment. The mass per unit length of each segment is as follows: \(\rho_{A B}\) = 12 kg/m,\(\rho_{B C}=8 \mathrm{~kg} / \mathrm{m}, \rho_{C D}=5 \mathrm{~kg} / \mathrm{m}, \text { and } \rho_{D E}=2 \mathrm{~kg} / \mathrm{m}\)

Determine the location \((\bar{y}\) of the centroidal axis \(\bar{x}-\bar{x}\) of the beam’s cross-sectional area. Neglect the size of the corner welds at A and B for the calculation

Locate the centroid \((\bar{x}, \bar{y})\) of the shaded area.

Locate the centroid \((\bar{y}\) for the beam’s cross-sectional area

Determine the location \((\bar{y}\) of the centroid C of the beam having the cross-sectional area shown.

Locate the centroid \((\bar{x}, \bar{y})\) of the shaded area

Determine the location \((\bar{y}\) of the centroid of the beam’s cross-sectional area. Neglect the size of the corner welds at A and B for the calculation.

Locate the centroid \((\bar{x}, \bar{y})\) of the shaded area

Determine the location \((\bar{x}, \bar{y})\) of the centroid C of the area.

Determine the location \((\bar{y}\) of the centroid C for a beam having the cross-sectional area shown. The beam is symmetric with respect to the y axis.

Locate the centroid \((\bar{y}\) of the cross-sectional area of the beam constructed from a channel and a plate. Assume all corners are square and neglect the size of the weld at A.

A triangular plate made of homogeneous material has a constant thickness that is very small. If it is folded over as shown, determine the location \((\bar{y}\) of the plate’s center of gravity G

A triangular plate made of homogeneous material has a constant thickness that is very small. If it is folded over as shown, determine the location \((\bar{z}\) of the plate’s center of gravity G.

Locate the center of mass \((\bar{z}\) of the forked level which is made from a homogeneous material and has the dimensions shown.

Determine the location \((\bar{x}\) of the centroid C of the shaded area that is part of a circle having a radius r.

A toy skyrocket consists of a solid conical top, \(\rho_{i}=600 \mathrm{~kg} / \mathrm{m}^{3}\) a hollow cylinder, \(\rho_{c}=400 \mathrm{~kg} / \mathrm{m}^{3}\), and a stick having a circular cross section, \(\rho_{s}=300 \mathrm{~kg} / \mathrm{m}^{3}\). Determine the length of the stick, x, so that the center of gravity G of the skyrocket is located along line aa.

Locate the centroid \((\bar{y}\) for the cross-sectional area of the angle.

Determine the location \((\bar{x}, \bar{y})\) of the center of gravity of the three-wheeler. The location of the center of gravity of each component and its weight are tabulated in the figure. If the three-wheeler is symmetrical with respect to the x–y plane, determine the normal reaction each of its wheels exerts on the ground

Locate the center of mass \((\bar{x}, \bar{y}, \bar{z})\) of the homogeneous block assembly.

The sheet metal part has the dimensions shown. Determine the location \((\bar{x}, \bar{y}, \bar{z})\) of its centroid.

The sheet metal part has a weight per unit area of 2 lb/\(ft^{2}\) and is supported by the smooth rod and the cord at C. If the cord is cut, the part will rotate about the y axis until it reaches equilibrium. Determine the equilibrium angle of tilt, measured downward from the negative x axis, that AD makes with the -x axis

The wooden table is made from a square board having a weight of 15 lb. Each of the legs weighs 2 lb and is 3 ft long. Determine how high its center of gravity is from the floor. Also, what is the angle, measured from the horizontal, through which its top surface can be tilted on two of its legs before it begins to overturn? Neglect the thickness of each leg.

The buoy is made from two homogeneous cones each having a radius of 1.5 ft. If h = 1.2 ft, find the distance z to the buoy’s center of gravity G.

The buoy is made from two homogeneous cones each having a radius of 1.5 ft. If it is required that the buoy’s center of gravity G be located at \((\bar{z}\) = 0.5 ft, determine the height h of the top cone

The assembly is made from a steel hemisphere, \(\rho_{s t}=7.80 \mathrm{Mg} / \mathrm{m}^{3}\), and an aluminum cylinder, \(\rho_{a l}=2.70 \mathrm{Mg} / \mathrm{m}^{3}\). Determine the mass center of the assembly if the height of the cylinder is h = 200 mm

The assembly is made from a steel hemisphere, \(\rho_{s t}=7.80 \mathrm{Mg} / \mathrm{m}^{3}\), and an aluminum cylinder, \(\rho_{a l}=2.70 \mathrm{Mg} / \mathrm{m}^{3}\). Determine the height h of the cylinder so that the mass center of the assembly is located at \((\bar{z}\) = 160 mm.

The car rests on four scales and in this position the scale readings of both the front and rear tires are shown by \(F_{A}\) and \(F_{B}\). When the rear wheels are elevated to a height of 3 ft above the front scales, the new readings of the front wheels are also recorded. Use this data to compute the location \((\bar{x}\) and \((\bar{y}\) to the center of gravity G of the car. The tires each have a diameter of 1.98 ft

Determine the distance h to which a 100-mm-diameter hole must be bored into the base of the cone so that the center of mass of the resulting shape is located at \((\bar{z}\) = 115 mm. The material has a density of 8 Mg/\(m^{3}\)

Determine the distance \((\bar{z}\) to the centroid of the shape that consists of a cone with a hole of height h = 50 mm bored into its base

Locate the center of mass \((\bar{z}\) of the assembly. The cylinder and the cone are made from materials having densities of 5 Mg/\(m^{3}\) and 9 Mg/\(m^{3}\), respectively

Major floor loadings in a shop are caused by the weights of the objects shown. Each force acts through its respective center of gravity G. Locate the center of gravity \((\bar{x}, \bar{y})\) of all these components

The assembly consists of a 20-in. wooden dowel rod and a tight-fitting steel collar. Determine the distance \((\bar{x}\) to its center of gravity if the specific weights of the materials are \(\gamma_{w}=150 \mathrm{lb} / \mathrm{ft}^{3} \text { and } \gamma_{s t}=490 \mathrm{lb} / \mathrm{ft}^{3}\). The radii of the dowel and collar are shown.

The composite plate is made from both steel (A) and brass (B) segments. Determine the mass and location \((\bar{x}, \bar{y}, \bar{z})\) of its mass center G. Take \(\rho_{s t}=7.85 \mathrm{Mg} / \mathrm{m}^{3}\) and \(\rho_{br}=8.74 \mathrm{Mg} / \mathrm{m}^{3}\).

Determine the volume of the silo which consists of a cylinder and hemispherical cap. Neglect the thickness of the plates

Determine the outside surface area of the storage tank

Determine the volume of the storage tank

Determine the surface area of the concrete seawall, excluding its bottom

A circular seawall is made of concrete. Determine the total weight of the wall if the concrete has a specific weight of \(\gamma_{c}=150 \mathrm{lb} / \mathrm{ft}^{3}\).

A ring is generated by rotating the quarter circular area about the x axis. Determine its volume.

A ring is generated by rotating the quarter circular area about the x axis. Determine its surface area.

Determine the volume of concrete needed to construct the curb.

Determine the surface area of the curb. Do not include the area of the ends in the calculation.

A ring is formed by rotating the area \(360^{\circ}\) about the \(\bar{x}-\bar{x}\) axes. Determine its surface area.

A ring is formed by rotating the area \(360^{\circ}\) about the \(\bar{x}-\bar{x}\) axes. Determine its volume.

The water-supply tank has a hemispherical bottom and cylindrical sides. Determine the weight of water in the tank when it is filled to the top at C. Take \(\gamma_{w}=62.4 \mathrm{lb} / \mathrm{ft}^{3}\).

Determine the number of gallons of paint needed to paint the outside surface of the water-supply tank, which consists of a hemispherical bottom, cylindrical sides, and conical top. Each gallon of paint can cover 250 \(ft^{2}\) .

Determine the surface area and the volume of the ring formed by rotating the square about the vertical axis

Determine the surface area of the ring. The cross section is circular as shown.

The heat exchanger radiates thermal energy at the rate of 2500 kJ/h for each square meter of its surface area. Determine how many joules (J) are radiated within a 5-hour period.

Determine the interior surface area of the brake piston. It consists of a full circular part. Its cross section is shown in the figure.

The suspension bunker is made from plates which are curved to the natural shape which a completely flexible membrane would take if subjected to a full load of coal. This curve may be approximated by a parabola, y = \(0.2x^{2}\). Determine the weight of coal which the bunker would contain when completely filled. Coal has a specific weight of \(\gamma=50 \mathrm{lb} / \mathrm{ft}^{3}\), and assume there is a 20% loss in volume due to air voids. Solve the problem by integration to determine the cross-sectional area of ABC; then use the second theorem of Pappus–Guldinus to find the volume

Determine the height h to which liquid should be poured into the cup so that it contacts three-fourths the surface area on the inside of the cup. Neglect the cup’s thickness for the calculation.

Determine the surface area of the roof of the structure if it is formed by rotating the parabola about the y axis

A steel wheel has a diameter of 840 mm and a cross section as shown in the figure. Determine the total mass of the wheel if \(\rho=5 \mathrm{Mg} / \mathrm{m}^{3}\)

Half the cross section of the steel housing is shown in the figure. There are six 10-mm-diameter bolt holes around its rim. Determine its mass. The density of steel is 7.85 Mg/\(m^{3}\). The housing is a full circular part.

The water tank has a paraboloid-shaped roof. If one liter of paint can cover 3 \(m^{2}\) of the tank, determine the number of liters required to coat the roof.

Determine the volume of material needed to make the casting.

Determine the height h to which liquid should be poured into the cup so that it contacts half the surface area on the inside of the cup. Neglect the cup’s thickness for the calculation

The pressure loading on the plate varies uniformly along each of its edges. Determine the magnitude of the resultant force and the coordinates \((\bar{x}, \bar{y})\) of the point where the line of action of the force intersects the plate. Hint: The equation defining the boundary of the load has the form p = ax + by + c, where the constants a, b, and c have to be determined.

The load over the plate varies linearly along the sides of the plate such that p = (12 - 6x + 4y) kPa. Determine the magnitude of the resultant force and the coordinates \((\bar{x}, \bar{y})\) of the point where the line of action of the force intersects the plate.

The load over the plate varies linearly along the sides of the plate such that \(p=\frac{2}{3}[x(4-y)]\) kPa. Determine the resultant force and its position \((\bar{x}, \bar{y})\) on the plate.

The rectangular plate is subjected to a distributed load over its entire surface. The load is defined by the expression \(p=p_{0} \sin (\pi x / a) \sin (\pi y / b) \text {, where } p_{0}\) represents the pressure acting at the center of the plate. Determine the magnitude and location of the resultant force acting on the plate

A wind loading creates a positive pressure on one side of the chimney and a negative (suction) pressure on the other side, as shown. If this pressure loading acts uniformly along the chimney’s length, determine the magnitude of the resultant force created by the wind.

When the tide water A subsides, the tide gate automatically swings open to drain the marsh B. For the condition of high tide shown, determine the horizontal reactions developed at the hinge C and stop block D. The length of the gate is 6 m and its height is 4 m. \(\rho_{w}=1.0 \mathrm{Mg} / \mathrm{m}^{3}\).

The tank is filled with water to a depth of d = 4 m. Determine the resultant force the water exerts on side A and side B of the tank. If oil instead of water is placed in the tank, to what depth d should it reach so that it creates the same resultant forces? \(\rho_{o}=900 \mathrm{~kg} / \mathrm{m}^{3} \text { and } \rho_{w}=1000 \mathrm{~kg} / \mathrm{m}^{3}\).

The concrete “gravity” dam is held in place by its own weight. If the density of concrete is \(\rho_{c}=2.5 \mathrm{Mg} / \mathrm{m}^{3}\), and water has a density of \(\rho_{w}=1.0 \mathrm{Mg} / \mathrm{m}^{3}\), determine the smallest dimension d that will prevent the dam from overturning about its end A.

The factor of safety for tipping of the concrete dam is defined as the ratio of the stabilizing moment due to the dam’s weight divided by the overturning moment about O due to the water pressure. Determine this factor if the concrete has a density of \(\rho_{conc}=2.5 \mathrm{Mg} / \mathrm{m}^{3}\) and for water \(\rho_{w}=1.0 \mathrm{Mg} / \mathrm{m}^{3}\).

The concrete dam in the shape of a quarter circle. Determine the magnitude of the resultant hydrostatic force that acts on the dam per meter of length. The density of water is \(\rho_{w}=1.0 \mathrm{Mg} / \mathrm{m}^{3}\).

The tank is used to store a liquid having a density of 80 lb/\(ft^{3}\). If it is filled to the top, determine the magnitude of force the liquid exerts on each of its two sides ABDC and BDFE.

The parabolic plate is subjected to a fluid pressure that varies linearly from 0 at its top to 100 lb/ft at its bottom B. Determine the magnitude of the resultant force and its location on the plate

The 2-m-wide rectangular gate is pinned at its center A and is prevented from rotating by the block at B. Determine the reactions at these supports due to hydrostatic pressure. \(\rho_{w}=1.0 \mathrm{Mg} / \mathrm{m}^{3}\)

The tank is filled with a liquid that has a density of 900 kg/\(m^{3}\) . Determine the resultant force that it exerts on the elliptical end plate, and the location of the center of pressure, measured from the x axis

Determine the magnitude of the resultant force acting on the gate ABC due to hydrostatic pressure. The gate has a width of 1.5 m. \(\rho_{w}=1.0 \mathrm{Mg} / \mathrm{m}^{3}\).

The semicircular drainage pipe is filled with water. Determine the resultant horizontal and vertical force components that the water exerts on the side AB of the pipe per foot of pipe length; \(\gamma_{w}=62.4 \mathrm{lb} / \mathrm{ft}^{3}\).