Problem 9.194 y Determine the moment of inertia and the radius of 15 mm x y z 40 mm 50 mm 60 mm 45 mm 38 mm Determine the moment of inertia and the radius of gyration of the steel machine element shown with respect to the x axis. (The density of steel is 7850 kg/m3.)

Solving Problems on Your Own 15 mm x y z 40 mm 50 mm 60 mm 45 mm 38 mm Solving Problems on Your Own Determine the moment of inertia and the radius of gyration of the steel machine element shown with respect to the x axis. (The density of steel is 7850 kg/m3.) Problem 9.194 1. Compute the mass moments of inertia of a composite body with respect to a given axis. 1a. Divide the body into sections. The sections should have a simple shape for which the centroid and moments of inertia can be easily determined (e.g. from Fig. 9.28 in the book).

Solving Problems on Your Own 15 mm x y z 40 mm 50 mm 60 mm 45 mm 38 mm Solving Problems on Your Own Determine the moment of inertia and the radius of gyration of the steel machine element shown with respect to the x axis. (The density of steel is 7850 kg/m3.) Problem 9.194 1b. Compute the mass moment of inertia of each section. The moment of inertia of a section with respect to the given axis is determined by using the parallel-axis theorem: I = I + m d2 Where I is the moment of inertia of the section about its own centroidal axis, I is the moment of inertia of the section about the given axis, d is the distance between the two axes, and m is the section’s mass.

k = I m Problem 9.194 Solving Problems on Your Own y 15 mm x y z 40 mm 50 mm 60 mm 45 mm 38 mm Solving Problems on Your Own Determine the moment of inertia and the radius of gyration of the steel machine element shown with respect to the x axis. (The density of steel is 7850 kg/m3.) Problem 9.194 1c. Compute the mass moment of inertia of the whole body. The moment of inertia of the whole body is determined by adding the moments of inertia of all the sections. 2. Compute the radius of gyration. The radius of gyration k of a body is defined by: k = I m

Divide the body into sections. 15 mm x y z 40 mm 50 mm 60 mm 45 mm 38 mm Problem 9.194 Solution Divide the body into sections. x y z x y z x z y

m = r V V = (0.15 m)(0.09 m)(0.03 m) V = 4.05 x 10-4 m3 y z 40 mm 50 mm 60 mm 45 mm 38 mm Problem 9.194 Solution Compute the mass moment of inertia of each section. Rectangular prism: m = r V V = (0.15 m)(0.09 m)(0.03 m) V = 4.05 x 10-4 m3 m = (7850 kg/m3)(4.05x10-4 m3) m = 3.18 kg x y z (Ix) = ( Ix’) + m d2 (Ix) = (3.18 kg)[(0.15 m)2+(0.03 m)2] + (3.18 kg) (0.075 m)2 (Ix) = 2.408 x 10-2 kg . m2 1 12 x’ 75 mm

15 mm x y z 40 mm 50 mm 60 mm 45 mm 38 mm Problem 9.194 Solution Semicircular cylinder: m = r V V = p (0.045 m)2 (0.04 m) V = 1.27 x 10-4 m3 m = (7850 kg/m3)(1.27x10-4 m3) m = 1.0 kg 1 2 x y z x’ 130 mm 15 mm x’’ 4r 3p 4.45 = 19.1 mm = (centroidal axis) C 40 mm

(1.0 kg) [3 (0.045 m)2 + (0.04 m)2] = Ix’’ + (1.0 kg)(0.0191 m)2 Problem 9.194 Solution x y z x’ 130 mm 15 mm x’’ 4r 3p 4.45 = 19.1 mm = (centroidal axis) C 40 mm Ix’ = Ix’’ + m d2 (1.0 kg) [3 (0.045 m)2 + (0.04 m)2] = Ix’’ + (1.0 kg)(0.0191 m)2 Ix’’ = 27.477 x 10-5 kg . m2 Ix = ( Ix’’) + m d2 = 27.477 x 10-5 kg . m2 + (1.0 kg)[(0.13 m)2 + (0.015 m + 0.0191 m)2] Ix = 18.34 x 10-3 kg . m2 1 12

15 mm x y z 40 mm 50 mm 60 mm 45 mm 38 mm Problem 9.194 Solution Circular cylindrical hole: m = r V V = p (0.038 m)2 (0.03 m) V = 1.361 x 10-4 m3 m = (7850 kg/m3)(1.36x10-4 m3) m = 1.068 kg (Ix) = ( Ix’) + m d2 (Ix) = (1.07 kg) [ 3 (0.038 m)2 + (0.03 m)2 ] + (1.07 kg) (0.06 m)2 (Ix) = 4.31 x 10-3 kg . m2 x z y x’ 60 mm 1 12

15 mm x y z 40 mm 50 mm 60 mm 45 mm 38 mm Problem 9.194 Solution Compute the mass moment of inertia of the whole body. For the entire body, adding the values obtained: Ix = 2.408 x 10-2 + 1.834 x 10-2 - 4.31 x 10-3 Ix = 3.81 x 10-2 kg . m2 Compute the radius of gyration. The mass of the entire body: m = 3.18 + 1.0 - 1.07 = 3.11 kg Radius of gyration: k = = k = 0.1107 m I m 3.81 x 10-2 kg . m2 3.11 kg