Solved: A triangular plate made of homogeneous material

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

Locate the center of gravity \(\bar{x}\) of the homogeneous rod bent in the form of a semicircular arc. The rod has a weight per unit length of 0.5 lb/ft. Also, determine the horizontal reaction at the smooth support B and the x and y components of reaction at the pin A.

Locate the distance \(\bar{x}\) to the center of gravity of the homogeneous rod bent into the parabolic shape. If the rod has a weight per unit length of 0.5 lb/ft, determine the reactions at the fixed support O.

Locate the distance \(\bar{y}\) to the center of gravity of the homogeneous rod bent into the parabolic shape.

Locate the centroid \((\bar{x},\ \bar{y})\) of the uniform rod. Evaluate the integrals using a numerical method.

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

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

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

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

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

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

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

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

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

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

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

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

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

If the density at any point in the quarter circular plate is defined by \(\rho = \rho_0 x y\), where \(\rho_0\) is a constant, determine the mass and locate the center of mass \((\bar{x},\ \bar{y})\) of the plate. The plate has a thickness t.

Determine the location \(\bar{r}\) of the centroid C of the cardioid, \(r = a(1 - \cos \theta)\).

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

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

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

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

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

Locate the centroid of the ellipsoid of revolution.

Locate the centroid \(\bar{z}\) of the right-elliptical 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 of the quarter-cone.

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

The king’s chamber of the Great Pyramid of Giza is located at its centroid. Assuming the pyramid to be a solid, prove that this point is at \(\bar{z} = \frac{1}{4}\ h\), Suggestion: Use a rectangular differential plate element having a thickness dz and area (2x)(2y).

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.

Locate the centroid for the wire. Neglect the thickness of the material and slight bends at the corners.

Locate the centroid \((\bar{x},\ \bar{y})\) of the cross section. All the dimensions are measured to the centerline thickness of each thin 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.

The three members of the frame each have a weight per unit length of 4 lb/ft. Locate the position \((\bar{x},\ \bar{y})\) of the center of gravity. Neglect the size of the pins at the joints and the thickness of the members. Also, calculate the reactions at the fixed support A.

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 \(\rho_s = 7.85\ Mg/m^3\), and for C, \(\rho_{al} = 2.71\ Mg/m^3\), determine the location \(\bar{x}\) of the center of mass. Neglect the size of the bolts.

To determine the location of the center of gravity of the automobile it is first placed in a level position, with the two wheels on one side resting on the scale platform P. In this position the scale records a reading of \(W_1\). Then, one side is elevated to a convenient height c as shown. The new reading on the scale is \(W_2\). If the automobile has a total weight of W, determine the location of its center of gravity \(G(\bar{x},\ \bar{y})\).

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 \((\bar{x},\ \bar{y})\) for the angle’s cross sectional area.

Locate the centroid \(\bar{y}\) of the channel’s cross sectional area.

Locate the centroid \((\bar{x},\ \bar{y})\) of the member’s cross sectional area.

Locate the centroid \(\bar{y}\) of the bulb-tee cross section.

Determine the location \((\bar{x},\ \bar{y})\) of the centroid C of the cross-sectional area for the structural member constructed from two equal-sized channels welded together as shown. Assume all corners are square. Neglect the size of the welds.

Locate the centroid \(\bar{y}\) of the concrete beam having the tapered cross section shown.

Locate the centroid \(\bar{y}\) of the beam’s cross-section built up from a channel and a wide-flange beam.

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.

An aluminum strut has a cross section referred to as a deep hat. Locate the centroid \(\bar{y}\) of its area. Each segment has a thickness of 10 mm.

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

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

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

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.

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.

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.

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 gravity of the two-block assembly. The specific weights of the materials A and B are \(\gamma_A = 150\ lb/ft^3\) and \(\gamma_B = 400\ lb/ft^3\), respectively.

Locate the center of mass of the block. Materials 1, 2, and 3 have densities of \(2.70\ Mg/m^3\), \(5.70\ Mg/m^3\), and \(7.80\ Mg/m^3\), respectively.

Locate the centroid \(\bar{z}\) of the homogeneous solid formed by boring a hemispherical hole into the cylinder that is capped with a cone.

Locate the center of mass \(\bar{z}\) of the solid formed by boring a hemispherical hole into a cylinder that is capped with a cone. The cone and cylinder are made of materials having densities of \(7.80\ Mg/m^3\) and \(2.70\ Mg/m^3\), respectively.

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.

The buoy is made from two homogeneous cones each having a radius of 1.5 ft. If h = 1.2 ft, find the distance \(\bar{z}\) 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.

Locate the center of mass \(\bar{z}\) of the assembly. The assembly consists of a cylindrical center core, A, having a density of \(7.90\ Mg/m^3\), and a cylindrical outer part, B, and a cone cap, C, each having a density of \(2.70\ Mg/m^3\).

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 \(\gamma_w = 150\ lb/ft^3\) and \(\gamma_{st} = 490\ lb/ft^3\). The radii of the dowel and collar are shown.

A hole having a radius r is to be drilled in the center of the homogeneous block. Determine the depth h of the hole so that the center of gravity G is as low as possible.

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.

Determine the outside surface area of the storage tank.

Determine the volume of the storage tank.

Determine the outside surface area of the hopper.

The hopper is filled to its top with coal. Determine the volume of coal if the voids (air space) are 35 percent of the volume of the hopper.

The rim of a flywheel has the cross section A–A shown. Determine the volume of material needed for its construction.

Determine the surface area of the concrete sea wall, excluding its bottom.

A circular sea wall is made of concrete. Determine the total weight of the wall if the concrete has a specific weight of \(\gamma_c = 150\ lb/ft^3\).

The process tank is used to store liquids during manufacturing. Estimate the outside surface area of the tank. The tank has a flat top and the plates from which the tank is made have negligible thickness.

The process tank is used to store liquids during manufacturing. Estimate the volume of the tank. The tank has a flat top and the plates from which the tank is made have negligible thickness.

The V-belt has an inner radius of 6 in. and a cross sectional area as shown. Determine the outside surface area of the belt.

A V-belt has an inner radius of 6 in., and a cross sectional area as shown. Determine the volume of material used in making the V-belt.

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 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 \(\gamma_w = 62.4\ lb/ft^3\).

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 \(250\ ft^2\).

The full circular aluminum housing is used in an automotive brake system. The cross section is shown in the figure. Determine its weight if aluminum has a specific weight of \(169\ lb/ft^3\).

Determine the volume of an ellipsoid formed by revolving the shaded area about the x axis using the second theorem of Pappus–Guldinus. The area and centroid y of the shaded area should first be obtained by using integration.

Using integration, determine both the area and the centroidal distance \(\bar{x}\) of the shaded area. Then, using the second theorem of Pappus–Guldinus, determine the volume of the solid generated by revolving the area about the y axis.

Determine the height h to which liquid should be poured into the conical cup so that it contacts half the surface area on the inside of the cup.

Determine the volume of the solid formed by revolving the shaded area about the u–u axis using the second theorem of Pappus–Guldinus. The area and centroid of the area should first be obtained by using integration.

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 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.

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 \(\rho = 5\ Mg/m^3\).

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

The pressure loading on the plate is described by the function \(\rho=10[6/(x + 1)+8]\ lb/ft^2\). 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 loading acting on a square plate is represented by a parabolic pressure distribution. 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. Also, what are the reactions at the rollers B and C and the ball-and-socket joint A? Neglect the weight of 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 \(\rho=\rho_0 \sin\ (\pi x / a)\ \sin\ (\pi y/b)\), where \(\rho_0\) represents the pressure acting at the center of the plate. Determine the magnitude and location of the resultant force acting on the plate.

The wind blows uniformly on the front surface of the metal building with a pressure of \(30\ lb/ft^2\). Determine the resultant force it exerts on the surface and the position of this resultant.

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? \(\rho_o = 900\ kg/m^3\) and \(\rho_w = 1000\ kg/m^3\).

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. \(\rho_w = 1.0\ Mg/m^3\).

Determine the resultant horizontal and vertical force components that the water exerts on the side of the dam. The dam is 25 ft long and \(\gamma_w = 62.4\ lb/ft^3\).

Determine the magnitude of the resultant hydrostatic force acting on the dam and its location, measured from the top surface of the water. The width of the dam is 8 m; \(\rho_w = 1.0\ Mg/m^3\).

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 \(\rho_w = 1\ Mg/m^3\).

The storage tank contains oil having a specific weight of \(\gamma_o = 56\ lb/ft^3\). If the tank is 6 ft wide, calculate the resultant force acting on the inclined side BC of the tank, caused by the oil, and specify its location along BC, measured from B. Also compute the total resultant force acting on the bottom of the tank.

The tank is filled to the top (y = 0.5 m) with water having a density of \(\rho_w = 1.0\ Mg/m^3\). Determine the resultant force of the water pressure acting on the flat end plate C of the tank, and its location, measured from the top of the tank.

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.

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; \(\gamma_w = 62.4\ lb/ft^3\).

The semicircular tunnel passes under a river that is 9 m deep. Determine the vertical resultant hydrostatic force acting per meter of length along the length of the tunnel. The tunnel is 6 m wide; \(\rho_w = 1.0\ Mg/m^3\).

The arched surface AB is shaped in the form of a quarter circle. If it is 8 m long, determine the horizontal and vertical components of the resultant force caused by the water acting on the surface; \(\rho_w = 1.0\ Mg/m^3\).