Determine the products of inertia about the coordinate axes for the thin plate of mass m

Use the mass element where is the mass per unit length, and determine the mass moments of inertia and of the homogeneous slender rod of mass and length l. Problem B/1

In order to better appreciate the greater ease of integration with lower-order elements, determine the mass moment of inertia of the homogeneous thin plate by using the square element (a) and then by using the rectangular element (b). The mass of the plate is m. Then by inspection state and finally, determine Problem B/2

Determine the mass moments of inertia about the x-, axes of the slender rod of length L and mass m which makes an angle with the axis as shown.

Determine the mass moment of inertia of the uniform thin triangular plate of mass m about the x-axis. Also determine the radius of gyration about the x-axis. By analogy state . Then determine Problem B/4

Calculate the moment of inertia of the tapered steel rod of circular cross section about an axis normal to the rod through O. Note that the rod diameter is small compared with its length. Problem B/5

Determine the mass moment of inertia of the uniform thin equilateral triangular plate of mass m about the x-axis. Also determine the corresponding radius of gyration. Problem B/6

Determine the mass moment of inertia about the y-axis for the equilateral triangular plate of the previous problem. Also determine its radius of gyration about the y-axis.

Determine the mass moments of inertia of the thin parabolic plate of mass m about the and axes. Problem B/8

Determine the mass moment of inertia of the uniform thin parabolic plate of mass m about the axis. State the corresponding radius of gyration.

Determine the mass of inertia about the axis for the parabolic plate of the previous problem. State the radius of gyration about the axis.

Calculate the moment of the homogeneous rightcircular cone of mass m, base radius r, and altitude h about the cone axis x and about the axis through its vertex. Problem B/11

Determine the mass moment of inertia of the uniform thin elliptical plate (mass m) about the axis. Then, by analogy, state the expression for Finally, determine Problem B/12

Determine the mass moment of inertia of the homogeneous solid of revolution of mass m about the axis. Problem B/13

Determine the mass moment of inertia of the homogeneous solid of revolution of the previous problem about the and axes.

Determine the mass moment of inertia about the x-axis for the uniform thin plate of mass m shown.

Determine the radius of gyration about the z-axis of the paraboloid of revolution shown. The mass of the homogenous body is m. Problem B/16

Determine the moment of inertia about the axis for the paraboloid of revolution of Prob. B/16

Determine the mass moment of inertia about the axis of the solid spherical segment of mass m. Problem B/18

Determine the moment of inertia about the axis of the homogeneous solid semiellipsoid of revolution having mass m. Problem B/19

Determine by integration the moment of inertia of the half-cylindrical shell of mass m about the axis The thickness of the shell is small compared with r. Problem B/21

Determine the moment of inertia about the generating axis of a complete ring (torus) of mass m having a circular section with the dimensions shown in the sectional view.

The plane area shown in the top portion of the figure is rotated about the axis to form the body of revolution of mass m shown in the lower portion of the figure. Determine the mass moment of inertia of the body about the axis. Problem B/23

Determine for the homogen eous body of revolution of the previous problem.

The thickness of the homogeneous triangular plate of mass m varies linearly with the distance from the vertex toward the base. The thickness a at the base is small compared with the other dimensions. Determine the moment of inertia of the plate about the axis along the centerline of the base. Problem B/25

Determine the moment of inertia of the triangular plate described in Prob. B/25 about the axis

Determine the moment of inertia of the triangular plate described in Prob. B/25 about the axis

Determine the moment of inertia, about the generating axis, of the hollow circular tube of mass m obtained by revolving the thin ring shown in the sectional view completely around the generating axis. Problem B/27

Determine the moments of inertia of the half-spherical shell with respect to the and axes. The mass of the shell is m, and its thickness is negligible compared with the radius r.

Determine the moment of inertia of the one-quartercylindrical shell of mass m about the axis. The thickness of the shell is small compared with r. Problem B/29

A shell of mass m is obtained by revolving the quarter-circular section about the axis. If the thickness of the shell is small compared with a and if a/3, determine the radius of gyration of the shell about the axis. Problem B/30

The two small spheres of mass m each are connected by the light rigid rod which lies in the plane. Determine the mass moments of inertia of the assembly about the and axes. Problem B/31

State without calculation the moment of inertia about the axis of the thin conical shell of mass m and radius r from the results of Sample Problem applied to a circular disk. Observe the radial distribution of mass by viewing the cone along the z-axis. Problem B/32

The moment of inertia of a solid homogeneous cylinder of radius r about an axis parallel to the central axis of the cylinder may be obtained approximately by multiplying the mass of the cylinder by the square of the distance d between the two axes. What percentage error e results if (a) and ( d 10r b) ?

Every slender rod has a finite radius r. Refer to Table and derive an expression for the percentage error e which results if one neglects the radius r of a homogeneous solid cylindrical rod of length l when calculating its moment of inertia . Evaluate your expression for the ratios Problem B/34

Calculate the mass moment of inertia about the axis O-O for the uniform 10-in. block of steel with crosssection dimensions of 6 and 8 in. Problem B/35

Determine for the cylinder with a centered circular hole. The mass of the body is m. Problem B/36

Determine the mass moment of inertia about the axis for the right-circular cylinder with a central longitudinal hole. Problem B/37

Determine the moment of inertia of the mallet about the axis. The density of the wooden handle is and that of the soft-metal head is The longitudinal axis of the cylindrical head is normal to the axis. State any assumptions. Problem B/38 300 mm 40 mm 3

Determine the moment of inertia of the half-ring of mass m about its diametral axis a-a and about axis b-b through the midpoint of the arc normal to the plane of the ring. The radius of the circular cross section is small compared with r. Problem B/39

The semicircular disk weighs 5 lb, and its small thickness may be neglected compared with its 10-in. radius. Compute the moments of inertia of the disk about the and axes. Problem B/40

A 6-in. steel cube is cut along its diagonal plane. Calculate the moment of inertia of the resulting prism about the edge . Problem B/41 x 6

Determine the length L of each of the slender rods of mass which must be centrally attached to the faces of the thin homogeneous disk of mass m in order to make the mass moments of inertia of the unit about the and axes equal. Problem B/42

A badminton racket is constructed of uniform slender rods bent into the shape shown. Neglect the strings and the built-up wooden grip and estimate the mass moment of inertia about the axis through O, which is the location of the players hand. The mass per unit length of the rod material is Problem B/43

As a sorting-machine part, the steel half-cylinder is subject to rapid angular acceleration and deceleration about the y-axis, and its moment of inertia about this axis is required in the design of the machine. Calculate Use Tables and as needed. Problem B/44

Calculate the moment of inertia of the steel control wheel, shown in section, about its central axis. There are eight spokes, each of which has a constant cross-sectional area of 200 . What percent n of the total moment of inertia is contributed by the outer rim? Problem B/45 Dimensions in millimeters 50 100 300 400 120 200 mm2 75

The uniform rod of length 4b and mass m is bent into the shape shown. The diameter of the rod is small compared with its length. Determine the moments of inertia of the rod about the three coordinate axes. Problem B/46

Calculate the moment of inertia of the solid steel semicylinder about the axis and about the parallel axis. (See Table for the density of steel.) Problem B/47 x0 60 mm 100 mm 60 mm x0 x x

The clock pendulum consists of the slender rod of length l and mass m and the bob of mass 7m. Neglect the effects of the radius of the bob and determine in terms of the bob position x. Calculate the ratio R of evaluated for to evaluated for . Problem B/48

Determine the moment of inertia about the axis of the portion of the homogeneous sphere shown. The mass of the sphere portion is m. Problem B/49

A square plate with a quarter-circular sector removed has a net mass m. Determine its moment of inertia about axis normal to the plane of the plate.

Determine the moments of inertia about the tangent axis for the full ring of mass and the half-ring of mass Problem B/51

The slender metal rods are welded together in the configuration shown. Each 6-in. segment weighs 0.30 Compute the moment of inertia of the assembly about the axis.

The welded assembly shown is made from a steel rod which weighs 0.667 per foot of length. Calculate the moment of inertia of the assembly about the axis.

The machine element is made of steel and is designed to rotate about axis O-O. Calculate its radius of gyration about this axis. Problem B/54

Determine for the cone frustum, which has base radii and and mass m.

By direct integration, determine the moment of inertia about the axis of the thin semicircular disk of mass m and radius R inclined at an angle from the plane

Determine the products of inertia about the coordinate axes for the unit which consists of three small spheres, each of mass m, connected by the light but rigid slender rods. Problem B/57

Determine the products of inertia about the coordinate axes for the unit which consists of four small particles, each of mass m, connected by the light but rigid slender rods. Problem B/58 z x y

Determine the products of inertia of the uniform slender rod of mass m about the coordinate axes shown. Problem B/59

Determine the products of inertia about the coordinate axes for the thin plate of mass m which has the shape of a circular sector of radius a and angle  as shown.

Determine the products of inertia about the coordinate axes for the thin square plate with two circular holes. The mass of the plate material per unit area is .

The slender rod of mass m is formed into a quartercircular arc of radius r. Determine the products of inertia of the rod with respect to the given axes

The uniform rectangular block weighs 50 lb. Calculate its products of inertia about the coordinate axes shown.

Determine the product of inertia Ixy for the slender rod of mass m. Problem B/64

The semicircular disk of mass m and radius R, inclined at an angle  from the X-y plane, of Prob. B/56 is repeated here. By the methods of this article, determine the moment of inertia about the Z-axis.

Determine the products of inertia for the rod of Prob. B/46, repeated here. Problem B/66 z b x O y b b b

The S-shaped piece is formed from a rod of diameter d and bent into the two semicircular shapes. Determine the products of inertia for the rod, for which d is small compared with r. Problem B/67

Determine the three products of inertia with respect to the given axes for the uniform rectangular plate of mass m. Problem B/68

For the slender rod of mass m bent into the configuration shown, determine its products of inertia Ixy, Ixz, and Iyz.

Determine the moment of inertia of the solid cube of mass m about the diagonal axis A-A through opposite corners. Problem B/70

The steel plate with two right-angle bends and a central hole has a thickness of 15 mm. Calculate its moment of inertia about the diagonal axis through the corners A and B. Problem B/71

Prove that the moment of inertia of the rigid assembly of three identical balls, each of mass m and radius r, has the same value for all axes through O. Neglect the mass of the connecting rods. Problem B/72 m z y x b b b m O m

Each sphere of mass m has a diameter which is small compared with the dimension b. Neglect the mass of the connecting struts and determine the principal moments of inertia of the assembly with respect to the coordinates shown. Determine also the direction cosines of the axis of maximum moment of inertia. Problem B/73

Determine the moment of inertia I about axis O-M for the uniform slender rod bent into the shape shown. Plot I versus  from  0 to  90 and determine the minimum value of I and the angle which its axis makes with the x-direction. (Note: Because the analysis does not involve the z-coordinate, the expressions developed for area moments of inertia, Eqs. A/9, A/10, and A/11 in Appendix A of Vol. 1 Statics, may be utilized for this problem in place of the three-dimensional relations of Appendix B.) The rod has a mass per unit length. Problem B/74

The assembly of three small spheres connected by light rigid bars of Prob. B/57 is repeated here. Determine the principal moments of inertia and the direction cosines associated with the axis of maximum moment of inertia. Problem B/75

The bent rod of Probs. B/46 and B/66 is repeated here. Its mass is m, and its diameter is small compared with its length. Determine the principal moments of inertia of the rod about the origin O. Also find the direction cosines for the axis of minimum moment of inertia. Problem B/76

The thin plate has a mass per unit area and is formed into the shape shown. Determine the principal moments of inertia of the plate about axes through O. Problem B/77

The slender rod has a mass per unit length and is formed into the shape shown. Determine the principal moments of inertia about axes through O and calculate the direction cosines of the axis of minimum moment of inertia. Problem B/78