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engineering
mechanical engineering
Vector Mechanics For Engineers Statics And Dynamics 8th Edition Ferdinand Beer, E. Russell Johnston, Jr., Elliot Eisenberg, William Clausen, David Mazurek, Phillip Cornwell - Solutions
To form a nonsymmetrical girder, two L3 Ã 3 Ã ¼-in. angles and two L6 Ã 4 Ã ½-in. angles are welded to a 0.8-in. steel plate as shown. Determine the moments of inertia of the combined section with respect to its centroidal x and y
Two L127 Ã 76 Ã 12.7-mm angles are welded to a 10-mm steel plate. Determine the distance b and the centroidal moments of inertia Ix and Iy of the combined section knowing that Iy= 3Ix.
A channel and an angle are welded to an a à 0.75-in. steel plate. Knowing that the centroidal y axis is located as shown, determine(a) The width a,(b) The moments of inertia with respect to the centroidal x and y axes.
The panel shown forms the end of a trough which is filled with water r to the line AA′. Referring to Sec. 9.2, determine the depth of the point of application of the resultant of the hydrostatic forces acting on the panel (the center of pressure).
The panel shown forms the end of a trough which is filled with water r to the line AA². Referring to Sec. 9.2, determine the depth of the point of application of the resultant of the hydrostatic forces acting on the panel (the center of pressure).
The panel shown forms the end of a trough which is filled with water r to the line AA². Referring to Sec. 9.2, determine the depth of the point of application of the resultant of the hydrostatic forces acting on the panel (the center of pressure).
The panel shown forms the end of a trough which is filled with water to the line AA′. Referring to Sec. 9.2, determine the depth of the point of application of the resultant of the hydrostatic forces acting on the panel (the center of pressure).
The cover for a 250 Ã 550-mm access hole in an oil storage tank is attached to the outside of the tank with four bolts as shown. Knowing that the density of the oil is 920 kg/m3 and that the center of the cover is located 3 m below the surface of the oil, determine the additional force
A vertical trapezoidal gate that is used as an automatic valve is held shut by two springs attached to hinges located along edge AB. Knowing that each spring exerts a couple of magnitude 8 kip ft, determine the depth d of water for which the gate will open.
Determine the x coordinate of the centroid of the volume shown.
Determine the x coordinate of the centroid of the volume shown; this volume was obtained by intersecting a circular cylinder with an oblique plane.
Show that the system of hydrostatic forces acting on a submerged plane area A can be reduced to a force P at the centroid C of the area and two couples. The force P is perpendicular to the area and is of magnitude P = γ Ay sin θ, where γ is the specific weight of the liquid, and
Show that the resultant of the hydrostatic forces acting on a submerged plane area A is a force P perpendicular to the area and of magnitude P = γAy sin θ = pA, where γ is the specific weight of the liquid and p is the pressure at the centroid C of the area.
Determine by direct integration the product of inertia of the given area with respect to the x and y axes.
Determine by direct integration the product of inertia of the given area with respect to the x and y axes.
Determine by direct integration the product of inertia of the given area with respect to the x and y axes.
Determine by direct integration the product of inertia of the given area with respect to the x and y axes.
Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.
Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.
Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.
Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.
Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.
Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.
Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.
Using the parallel-axis theorem, determine the product of inertia of the area shown with respect to the centroidal x and y axes.
Determine for the quarter ellipse of Prob. 9.67 the moments of inertia and the product of inertia with respect to new axes obtained by rotating the x and y axes about O(a) Through 45o counterclockwise,(b) Through 30o clockwise.
Determine the moments of inertia and the product of inertia of the area of Prob. 9.72 with respect to new centroidal axes obtained by rotating the x and y axes 45° clockwise.
Determine the moments of inertia and the product of inertia of the area of Prob. 9.73 with respect to new centroidal axes obtained by rotating the x and y axes through 30o clockwise.
Determine the moments of inertia and the product of inertia of the area of Prob. 9.75 with respect to new centroidal axes obtained by rotating the x and y axes through 60o counterclockwise.
Determine the moments of inertia and the product of inertia of the L76 Ã 51 Ã 6.4-mm angle cross section of Prob. 9.74 with respect to new centroidal axes obtained by rotating the x and y axes hrough 45o clockwise.
Determine the moments of inertia and the product of inertia of the L5 Ã 3 Ã ½ -in. angle cross section of Prob. 9.78 with respect to new centroidal axes obtained by rotating the x and y axes through 30o counterclockwise.
For the quarter ellipse of Prob. 9.67, determine the orientation of the principal axes at the origin and the corresponding values of the moments of inertia.
For the area indicated, determine the orientation of the principal axes at the origin and the corresponding values of the moments of inertia. Area of Prob. 9.72
For the area indicated, determine the orientation of the principal axes at the origin and the corresponding values of the moments of inertia. Area of Prob. 9.73
For the area indicated, determine the orientation of the principal axes at the origin and the corresponding values of the moments of inertia. Area of Prob. 9.75
For the angle cross section indicated, determine the orientation of the principal axes at the origin and the corresponding values of the moments of inertia. The L76 Ã 51 Ã 6.4-mm angle cross section of Prob. 9.74
For the angle cross section indicated, determine the orientation of the principal axes at the origin and the corresponding values of the moments of inertia. The L5 Ã 3 Ã 1/2-in.angle cross section of Prob. 9.78
Using Mohrs circle, determine for the quarter ellipse of Prob. 9.67 the moments of inertia and the product of inertia with respect to new axes obtained by rotating the x and y axes about O(a) Through 45o counterclockwise,(b) Through 30o clockwise.
Using Mohrs circle, determine the moments of inertia and the product of inertia of the area of Prob. 9.72 with respect to new centroidal axes obtained by rotating the x and y axes 45° clockwise.
Using Mohrs circle, determine the moments of inertia and the product of inertia of the area of Prob. 9.73 with respect to new centroidal axes obtained by rotating the x and y axes through 30o clockwise.
Using Mohrs circle, determine the moments of inertia and the product of inertia of the area of Prob. 9.75 with respect to new centroidal axes obtained by rotating the x and y axes through rough 60o counterclockwise.
Using Mohrs circle, determine the moments of inertia and the product of inertia of the L76 Ã 51 Ã 6.4-mm angle cross section of Prob. 9.74 with respect to new centroidal axes obtained by rotating the x and y axes through h 45o clockwise.
Using Mohrs circle, determine the moments of inertia and the product of inertia of the L5 Ã 3 Ã ½-in. angle cross section of Prob. 9.78 with respect to new centroidal axes obtained by rotating the x and y axes through h 30o counterclockwise.
For the quarter ellipse of Prob. 9.67, use Mohrs circle to determine the orientation of the principal axes at the origin and the corresponding values of the moments of inertia.
Using Mohrs circle, determine for the area indicated the orientation of the principal centroidal axes and the corresponding values of the moments of inertia. Area of Prob. 9.72
Using Mohrs circle, determine for the area indicated the orientation of the principal centroidal axes and the corresponding values of the moments of inertia. Area of Prob. 9.76
Using Mohrs circle, determine for the area indicated the orientation of the principal centroidal axes and the corresponding values of the moments of inertia. Area of Prob. 9.73
Using Mohrs circle, determine for the area indicated the orientation of the principal centroidal axes and the corresponding values of the moments of inertia. Area of Prob. 9.74
Using Mohrs circle, determine for the area indicated the orientation of the principal centroidal axes and the corresponding values of the moments of inertia. Area of Prob. 9.75
Using Mohrs circle, determine for the area indicated the orientation of the principal centroidal axes and the corresponding values of the moments of inertia. Area of Prob. 9.71
Using Mohrs circle, determine for the area indicated the orientation of the principal centroidal axes and the corresponding values of the moments of inertia. Area of Prob. 9.77 (The moments of inertia Ix and Iy of the area of Prob. 9.104 were determined in Prob. 9.43.)
The moments and product of inertia for an L102 × 76 × 6.4-mm angle cross section with respect to two rectangular axes x and y through C are, respectively, Ix = 0.166 x 106 mm4, Ix = 0.453 x 106 mm4, and Ixy < 0, with the minimum value of the moment of inertia of the area with respect to any axis
Using Mohrs circle, determine for the cross section of the rolled-steel angle shown the orientation of the principal centroidal axes and the corresponding values of the moments of inertia. (Properties of the cross sections are given in Fig. 9.13.)
Using Mohrs circle, determine for the cross section of the rolled-steel angle shown the orientation of the principal centroidal axes and the corresponding values of the moments of inertia. (Properties of the cross sections are given in Fig. 9.13.)
For a given area the moments of inertia with respect to two rectangular centroidal x and y axes are Ix = 640 in4 and Iy = 280 in4, respectively. Knowing that after rotating the x and y axes about the centroid 60° clockwise the product of inertia relative to the rotated axes is −180 in4, use
It is known that for a given area Iy = 300 in4 and Ixy = −125 in4, where the x and y axes are rectangular centroidal axes. If the axis corresponding to the maximum product of inertia is obtained by rotating the x axis 67.5o counterclockwise about C, use Mohr’s circle to determine (a) The
Using Mohr’s circle, show that for any regular polygon (such as a pentagon)(a) The moment of inertia with respect to every axis through the centroid is the same,(b) The product of inertia with respect to every pair of rectangular axes through the centroid is zero.
Using Mohr’s circle, prove that the expression Ix’ Iy’ −I2x'y' is independent of the orientation of the x′ and y′ axes, where Ix' , Iy′ , and Ix′y′ represent the moments and product of inertia, respectively, of a given area with respect to a pair of
Using the invariance property established in the preceding problem, express the product of inertia Ixy of an area A with respect to a pair of rectangular axes through O in terms of the moments of inertia Ix and Iy of A and the principal moments of inertia Imin and Imax of A about O. Use the formula
A thin semicircular plate has a radius a and a mass m. Determine the mass moment of inertia of the plate with respect to(a) The centroidal al axis BB′,(b) The centroidal axis CC′ that is perpendicular to the plate.
The thin circular ring shown was cut from a thin, uniform plate. Denoting the mass of the ring by m, determine its mass moment of inertia with respect to(a) The centroidal axis AA′ of the ring,(b) The centroidal axis CC′ that is perpendicular to the plane of the ring.
The quarter ring shown has a mass m and was cut from a thin, uniform plate. Knowing that r1= ½r2, determine the mass moment of inertia of the quarter ring with respect to(a) Axis AA²,(b) The centroidal axis CC² that is perpendicular to the plane of the quarter ring.
The spacer shown was cut from a thin, uniform plate. Denoting the mass of the component by m, determine its mass moment of inertia with respect to(a) The axis AA′,(b) The centroidal axis CC′ that is perpendicular to the plane of the component.
A thin plate of mass m is cut in the shape of an isosceles triangle of base b and height h. Determine the mass moments of inertia of the plate with respect to(a) The centroidal axes AA² and BB² in the plane of the plate,(b) The centroidal axis CC² perpendicular to
A thin plate of mass m is cut in the shape of an isosceles triangle of base b and height h. Determine the mass moments of inertia of the plate with respect to axes DD′ and EE′ parallel to the centroidal axes AA′ and BB′ , respectively, and located at a distance d from the
A thin plate of mass m has the trapezoidal shape shown. Determine the mass moment of inertia of the plate with respect to(a) The x axis,(b) The y axis.
A thin plate of mass m has the trapezoidal shape shown. Determine the mass moment of inertia of the plate with respect to(a) The centroidal axis CC′ that is perpendicular to the plate,(b) The axis AA′ which is parallel to the x axis and is located at a distance 1.5a from the plate.
The parabolic spandrel shown is revolved about the x axis to form a homogeneous solid of revolution of mass m, using direct integration, express the moment of inertia of the solid with respect to the x axis in terms of m and b.
Determine by direct integration the mass moment of inertia with respect to the z axis of the truncated right circular cone shown knowing that the radius of the base r1 = 2r2 and assuming that the cone has a uniform density and a mass m.
The area shown is revolved about the x axis to form a homogeneous solid of revolution of mass m. Determine by direct integration the mass moment of inertia of the solid with respect to(a) The x axis,(b) The y axis. Express your answers in terms of m and a.
Determine by direct integration the mass moment of inertia with respect to the x axis of the pyramid shown assuming that it has a uniform density and a mass m.
Determine by direct integration the mass moment of inertia with respect to the y axis of the pyramid shown assuming that it has a uniform density and a mass m.
Determine by direct integration the mass moment of inertia and the radius of gyration with respect to the y axis of the paraboloid shown assuming that it has a uniform density and a mass m.
A thin steel wire is bent into the shape shown. Denoting the mass ss per unit length of the wire by m′, determine by direct integration the mass moment of inertia of the wire with respect to each of the coordinate axes.
A thin triangular plate of mass m is welded along its base AB to a block as shown. Knowing that the plate forms an angle θ with the y axis, determine by direct integration the mass moment of inertia of the plate with respect to(a) The x axis,(b) The y axis,(c) The z axis.
Shown is the cross section of the wheel of a caster. Determine its mass moment of inertia and radius of gyration with respect to axis AA′. (The specific weight of bronze is 0.310 lb/in3; of steel, 0.284 lb/in3; and of hard rubber, 0.043 lb/in3.)
Shown is the cross section of an idler roller. Determine its mass moment of inertia and its radius of gyration with respect to the axis AA². (The density of bronze is 8580 kg/m3; of aluminum, 2770 kg/m3; and of neoprene, 1250 kg/m3.)
Knowing that the thin hemispherical shell shown is of mass m and thickness t, determine the mass moment of inertia and the radius of gyration of the shell with respect to the x axis.
For the homogeneous ring of density ρ shown, determine(a) The mass moment of inertia with respect to the axis BB′,(b) The value of a1 for which, given a2 and h, IBB′ is maximum,(c) The corresponding value of IBB'
The steel machine component shown is formed by machining a hemisphere into the base of a truncated cone. Knowing that the density of steel is 7850 kg/m3, determine the mass moment of inertia of the component with respect to the y axis.
After a period of use, one of the blades of a shredder has been worn to the shape shown and weighs 0.4 lb.Knowing that the mass moments of inertia of the blade with respect to the AA′ and BB′ axes are 0.6 Ã 10−3 lb ⋅ ft ⋅ s2 and 1.26 Ã
The cups and the arms of an anemometer are fabricated from a material of density ρ. Knowing that the mass moment of inertia of a thin, hemispherical shell of mass m and thickness t with respect to its centroidal axis GG′ is 5ma2/12, determine(a) The mass moment of inertia of the
A square hole is centered in and extends through the aluminum machine component shown. Determine(a) The value of a for which the mass moment of inertia of the component with respect to the axis AA², which bisects the top surface of the hole, is maximum,(b) The corresponding values of the
The machine component shown is fabricated from 0.08-in.-thick sheet steel. Knowing that the specific weight of steel is 490 lb/ft3, determine the mass moment of inertia of the component with respect to each of the coordinate axes.
A 3-mm-thick piece of sheet metal is cut and bent into the machine component shown. Knowing that the density of steel is 7850 kg/m3, determine the mass moment of inertia of the component with respect to each of the coordinate axes.
A 2-mm-thick piece of sheet steel is cut and bent into the machine component shown. Knowing that the density of steel is 7850 kg/m3, determine the mass moment of inertia of the component with respect to each of the coordinate axes.
A framing anchor is formed from 2-mm-thick galvanized steel. Determine the mass moment of inertia of the anchor with respect to each of the coordinate axes. (The density of galvanized steel is 7530 kg/m3.)
A 0.1-in.-thick piece of sheet steel is cut and bent into the machine component shown. Knowing that the specific weight of steel is 490 lb/ft3, determine the mass moment of inertia of the component with respect to each of the coordinate axes.
The piece of roof flashing shown is formed from sheet copper that is 0.8 mm thick. Knowing that the density of copper is 8940 kg/m3, determine the mass moment of inertia of the flashing with respect to each of the coordinate axes.
The machine element shown is fabricated from steel. Determine the mass moment of inertia of the assembly with respect to(a) The x axis,(b) The y axis,(c) The z axis. (The specific weight of steel is 0.284 lb/in3)
Determine the mass moment of inertia of the steel machine element shown with respect to the y axis. (The density of steel is 7850 kg/m3.)
Determine the mass moment of inertia of the steel machine element shown with respect to the z axis. (The density of steel is 7850 kg/m3.)
An aluminum casting has the shape shown. Knowing that the specific weight of aluminum is 0.100 lb/in3, determine the mass moment of inertia of the casting with respect to the z axis.
Determine the mass moment of inertia of the steel machine element shown with respect to(a) The x axis,(b) The y axis,(c) The z axis. (The specific weight of steel is 490 lb/ft3.)
Aluminum wire with a mass per unit length of 0.049 kg/m is used to form the circle and the straight members of the figure shown. Determine the mass moment of inertia of the assembly with respect to each of the coordinate axes.
The figure shown is formed of 3-mm-diameter steel wire. Knowing that the density of the steel l is 7850 kg/m3, determine the mass moment of inertia of the wire with respect to each of the coordinate axes.
A homogeneous wire with a weight per unit length of 0.041 lb/ft is used to form the figure shown. Determine the mass moment of inertia of the wire with respect to each of the coordinate axes.
Determine the mass products of inertia Ixy, Iyz, and Izx of the steel machine element shown. (The specific weight of steel is 490 lb/ft3.)
Determine the mass products of inertia Ixy, Iyz, and Izx of the aluminum casting shown. (The specific weight of aluminum is 0.100 lb/in3.)
Determine the mass products of inertia Ixy, Iyz, and Izx of the cast aluminum machine component shown. (The density of aluminum is 2700 kg/m3.)
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