Locate the centroid y of the channels crosssectional area,

Determine the moment of inertia of the shaded area about the x axis.

Determine the moment of inertia of the shaded area about the y axis.

Determine the moment of inertia of the area about the x axis.

Determine the moment of inertia of the area about the y axis.

Determine the moment of inertia of the area about the x axis.

Determine the moment of inertia of the area about the y axis.

Determine the moment of inertia of the area about the x axis.

Determine the moment of inertia of the area about the y axis.

Determine the moment of inertia of the area about the x axis.

Determine the moment of inertia of the area about the x axis. Solve the problem in two ways, using rectangular differential elements: (a) having a thickness dx and (b) having a thickness of dy.

Determine the moment of inertia of the area about the x axis.

Determine the moment of inertia of the area about the y axis.

Determine the moment of inertia of the area about the x axis. Solve the problem in two ways, using rectangular differential elements: (a) having a thickness of dx, and (b) having a thickness of dy.

Determine the moment of inertia of the area about the y axis. Solve the problem in two ways, using rectangular differential elements: (a) having a thickness of dx, and (b) having a thickness of dy.

Determine the moment of inertia of the shaded area about the y axis. Use Simpson’s rule to evaluate the integral.

Determine the moment of inertia of the shaded area about the x axis. Use Simpson’s rule to evaluate the integral.

Determine the moment of inertia of the shaded area about the x axis.

Determine the moment of inertia of the shaded area about the y axis.

Determine the moment of inertia of the shaded area about the x axis.

Determine the moment of inertia of the shaded area about the y axis.

Determine the moment of inertia of the shaded area about the x axis.

Determine the moment of inertia of the shaded area about the y axis.

Determine the moment of inertia of the shaded area about the x axis.

Determine the moment of inertia of the shaded area about the y axis.

Determine the moment of inertia of the composite area about the x axis.

Determine the moment of inertia of the composite area about the y axis.

Determine the radius of gyration \(k_x\) for the column’s cross-sectional area.

Determine the moment of inertia of the beam’s cross-sectional area about the x axis.

Locate the centroid \(\bar{y}\) of the channel’s cross sectional area, and then determine the moment of inertia with respect to the x’ axis passing through the centroid.

Determine the distance \(\bar{x}\) to the centroid of the beam’s cross-sectional area, then find the moment of inertia about the y’ axis.

Determine the moment of inertia of the beam’s cross-sectional area about the x’ axis.

Determine the moment of inertia of the shaded area about the x axis.

Determine the moment of inertia of the shaded area about the y axis.

Determine the moment of inertia of the beam’s cross-sectional area about the y axis.

Determine \(\bar{y}\), which locates the centroidal axis x’ for the cross-sectional area of the T-beam, and then find the moment of inertia about the x’ axis.

Determine the moment of inertia \(I_x\) of the shaded area about the x axis.

Determine the moment of inertia \(I_y\) of the shaded area about the y axis.

The beam is constructed from the two channels and two cover plates. If each channel has a cross-sectional area of \(A_c = 11.8\ in^2\) and a moment of inertia about a horizontal axis passing through its own centroid, \(C_c\), of \((\bar{I}_{\bar{x}})_{C_c} = 349\ in^4\), determine the moment of inertia of the beam about the x axis.

The beam is constructed from the two channels and two cover plates. If each channel has a cross-sectional area of \(A_c = 11.8\ in^2\) and a moment of inertia about a vertical axis passing through its own centroid, \(C_c\), of \((\bar{I}_{\bar{y}})_{C_c} = 9.23\ in^4\), determine the moment of inertia of the beam about the y axis.

Locate the centroid \(\bar{y}\) of the composite area, then determine the moment of inertia of this area about the centroidal x’ axis.

Determine the moment of inertia of the composite area about the centroidal y axis.

Determine the moment of inertia of the beam’s cross-sectional area about the x axis.

Determine the moment of inertia of the beam’s cross-sectional area about the y axis.

Determine the distance \(\bar{y}\) to the centroid C of the beam’s cross-sectional area and then compute the moment of inertia \(\bar{I}_{x’}\) about the x’ axis.

Determine the distance \(\bar{x}\) to the centroid C of the beam’s cross-sectional area and then compute the moment of inertia \(\bar{I}_{y’}\) about the y’ axis.

Determine the distance \(\bar{y}\) to the centroid for the beam’s cross-sectional area; then determine the moment of inertia about the x’ axis.

Determine the moment of inertia of the beam’s cross-sectional area about the y axis.

Determine the moment of inertia of the area about the x axis.

Determine the moment of inertia of the area about the y axis.

Locate the centroid \(\bar{y}\) of the cross section and determine the moment of inertia of the section about the x’ axis.

Determine the moment of inertia of the beam’s cross-sectional area with respect to the x’ centroidal axis. Neglect the size of all the rivet heads, R, for the calculation. Handbook values for the area, moment of inertia, and location of the centroid C of one of the angles are listed in the figure.

Determine the moment of inertia of the parallelogram about the x’ axis, which passes through the centroid C of the area.

Determine the moment of inertia of the parallelogram about the y’ axis, which passes through the centroid C of the area.

Determine the product of inertia of the thin strip of area with respect to the x and y axes. The strip is oriented at an angle \(\theta\) from the x axis. Assume that \(t\ \ll\ l\).

Determine the product of inertia of the shaded area with respect to the x and y axes.

Determine the product of inertia of the shaded area with respect to the x and y axes.

Determine the product of inertia of the shaded area with respect to the x and y axes.

Determine the product of inertia of the shaded area with respect to the x and y axes, and then use the parallel-axis theorem to find the product of inertia of the area with respect to the centroidal x’ and y’ axes.

Determine the product of inertia of the shaded area with respect to the x and y axes. Use Simpson’s rule to evaluate the integral.

Determine the product of inertia of the shaded area with respect to the x and y axes.

Determine the product of inertia of the parallelogram with respect to the x and y axes.

Determine the product of inertia of the shaded area with respect to the x and y axes.

Determine the product of inertia for the beam’s cross-sectional area with respect to the u and v axes.

Determine the moments of inertia of the shaded area with respect to the u and v axes.

Determine the product of inertia of the beam’s cross-sectional area with respect to the x and y axes that have their origin located at the centroid C.

Determine the product of inertia of the cross sectional area with respect to the x and y axes.

Determine the product of inertia for the angle with respect to the x and y axes passing through the centroid C. Assume all corners to be square.

Determine the distance \(\bar{y}\) to the centroid of the area and then calculate the moments of inertia \(I_u\) and \(I_v\) of the channel’s cross-sectional area. The u and v axes have their origin at the centroid C. For the calculation, assume all corners to be square.

Determine the moments of inertia \(I_u\) and \(I_v\) of the shaded area.

Determine the moments of inertia and the product of inertia of the beam’s cross sectional area with respect to the u and v axes.

Solve Prob. 10–70 using Mohr’s circle. Hint: Once the circle is established, rotate \(2 \theta = 60^{\circ}\) counterclockwise from the reference OA, then find the coordinates of the points that define the diameter of the circle.

Locate the centroid \(\bar{y}\) of the beam’s cross-sectional area and then determine the moments of inertia and the product of inertia of this area with respect to the u and v axes.

Solve Prob. 10–72 using Mohr’s circle. Hint: Once the circle is established, rotate \(2 \theta = 120^{\circ}\) counterclockwise from the reference OA, then find the coordinates of the points that define the diameter of the circle.

Determine the principal moments of inertia of the beam’s cross-sectional area about the principal axes that have their origin located at the centroid C. Use the equations developed in Section 10.7. For the calculation, assume all corners to be square.

Solve Prob. 10–74 using Mohr’s circle.

Determine the directions of the principal axes with origin located at point O, and the principal moments of inertia for the rectangular area about these axes.

Solve Prob. 10–76 using Mohr’s circle.

Determine the principal moments of inertia for the angle’s cross-sectional area with respect to a set of principal axes that have their origin located at the centroid C. Use the equation developed in Section 10.7. For the calculation, assume all corners to be square.

Solve Prob. 10–78 using Mohr’s circle.

Determine the directions of the principal axes with origin located at point O, and the principal moments of inertia of the area about these axes.

Solve Prob. 10–80 using Mohr’s circle.

The area of the cross section of an airplane wing has the following properties about the x and y axes passing through the centroid \(C: \bar{I}_x = 450\ in^4\), \(\bar{I}_y = 1730\ in^4\), \(\bar{I}_{xy} = 138\ in^4\). Determine the orientation of the principal axes and the principal moments of inertia.

Solve Prob. 10–82 using Mohr’s circle.

Determine the moment of inertia of the thin ring about the z axis. The ring has a mass m.

Determine the moment of inertia of the semiellipsoid with respect to the x axis and express the result in terms of the mass m of the semiellipsoid. The material has a constant density \(\rho\).

Determine the radius of gyration \(k_x\) of the body. The specific weight of the material is \(\gamma = 380\ lb/ft^3\).

Determine the radius of gyration \(k_x\) of the paraboloid. The density of the material is \(\rho = 5\ Mg/m^3\).

Determine the moment of inertia of the ellipsoid with respect to the x axis and express the result in terms of the mass m of the ellipsoid. The material has a constant density \(\rho\).

Determine the moment of inertia of the homogenous triangular prism with respect to the y axis. Express the result in terms of the mass m of the prism. Hint: For integration, use thin plate elements parallel to the x–y plane having a thickness of dz.

Determine the mass moment of inertia \(I_z\) of the solid formed by revolving the shaded area around the z axis. The density of the materials is \(\rho\). Express the result in terms of the mass m of the solid.

Determine the moment of inertia \(I_x\) of the sphere and express the result in terms of the total mass m of the sphere. The sphere has a constant density \(\rho\).

The concrete shape is formed by rotating the shaded area about the y axis. Determine the moment of inertia \(I_y\). The specific weight of concrete is \(\gamma = 150\ lb/ft^3\).

Determine the mass moment of inertia \(I_y\) of the solid formed by revolving the shaded area around the y axis. The density of the material is \(\rho\). Express the result in terms of the mass m of the solid.

Determine the mass moment of inertia \(I_y\) of the solid formed by revolving the shaded area around the y axis. The total mass of the solid is 1500 kg.

The slender rods have a weight of \(3\ lb/ft\). Determine the moment of inertia of the assembly about an axis perpendicular to the page and passing through point A.

The pendulum consists of a disk having a mass of 6 kg and slender rods AB and DC which have a mass of 2 kg/m. Determine the length L of DC so that the center of the mass is at the bearing O. What is the moment of inertia of the assembly about an axis perpendicular to the page and passing through point O?

The pendulum consists of the 3-kg slender rod and the 5-kg thin plate. Determine the location \(\bar{y}\) of the center of mass G of the pendulum; then find the mass moment of inertia of the pendulum about an axis perpendicular to the page and passing through G.

Determine the location \(\bar{y}\) of the center of mass G of the assembly and then calculate the moment of inertia about an axis perpendicular to the page and passing through G. The block has a mass of 3 kg and the mass of the semicylinder is 5 kg.

If the large ring, small ring and each of the spokes weigh 100 lb, 15 lb, and 20 lb, respectively, determine the mass moment of inertia of the wheel about an axis perpendicular to the page and passing through point A.

Determine the mass moment of inertia of the assembly about the z axis. The density of the material is \(7.85\ Mg/m^3\).

Determine the moment of inertia \(I_z\) of the frustum of the cone which has a conical depression. The material has a density of \(200\ kg/m^3\).

The pendulum consists of a plate having a weight of 12 lb and a slender rod having a weight of 4 lb. Determine the radius of gyration of the pendulum about an axis perpendicular to the page and passing through point O.

The slender rods have a weight of \(3\ lb/ft\). Determine the moment of inertia of the assembly about an axis perpendicular to the page and passing through point A.

Determine the moment of inertia \(I_z\) of the frustrum of the cone which has a conical depression. The material has a density of \(200\ kg/m^3\).

Determine the moment of inertia of the wire triangle about an axis perpendicular to the page and passing through point O. Also, locate the mass center G and determine the moment of inertia about an axis perpendicular to the page and passing through point G. The wire has a mass of 0.3 kg/m. Neglect the size of the ring at O.

The thin plate has a mass per unit area of \(10\ kg/m^2\). Determine its mass moment of inertia about the y axis.

The thin plate has a mass per unit area of \(10\ kg/m^2\). Determine its mass moment of inertia about the z axis.

Determine the moment of inertia of the wheel about the x axis that passes through the center of mass G. The material has a specific weight of \(\gamma = 90\ lb/ft^3\).

Determine the moment of inertia of the wheel about the x’ axis that passes through point O. The material has a specific weight of \(\gamma = 90\ lb/ft^3\).