Moment of Inertia Calculations in Engineering Mechanics - Problems and Solutions
The content covers problems related to calculating moments of inertia in engineering mechanics, specifically focusing on triangular areas, shaded areas, and circles. Detailed step-by-step solutions are provided for each problem, including determining moments of inertia about different axes and finding radii of gyration. Diagrams and formulas are utilized to explain the concepts thoroughly.
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STATICS (ENGINEERING MECHANICS-I) Area moment of Inertia-Problems October 7, 2024 1
Problem-1 Determine the moments of inertia of the triangular area about: (i) The base, (ii) The parallel axis passing through its centroid, (iii) The parallel axis passing through its vertex, 10/7/2024 2
Solution The i. moment of inertia about the base i.e. - axis : x h h 3 4 1 h y y y 2 = = = = 2 3 bdy I y dA y b bh x 3 4 12 h h 0 0 ii. The moment of inertia x about the centroidal axis I (a distance above h/3 the - axis) : 2 2 1 1 1 h h = = = 3 3 I I A bh bh bh x 3 12 2 3 36 iii . transfer A from the centroiuda axis l to the x'-axis through th vertex e gives : 2 2 2 1 1 2 1 h h = + = + = 3 3 I I A bh bh bh 3 36 2 3 4 10/7/2024 3
Problem-2 For the shaded area and the strip shown in the two figures, calculate the moment of Inertia and radius of gyration about the x-axis (i.e. Ix and rx) y y 3 m 3 m y = x2 / 12 y = x2 / 12 x x 6 m 6 m October 7, 2024 4
Solution y 3 m 3 3 0 0 = = = 1 2 / 6 ( ) 6 ( 2 y 3 dy ) A dA -x dy - y = x2 / 12 3 3 2 / y = = 2 6 2 3 6 m y - 3 2 / x 6 m 0 3 3 = = = 2 2 2 (6 - ) (6 - 12 ) I y dA y x dy y y dy x 0 0 3 3 3 7 2 / y y = = = 2 5/2 4 (6 - 2 3 ) 6 2 3 . 7 714 m I y y dy - x 3 7 2 / 0 0 Radius of gyration about the - axis : x . 7 714 I = = = . 1 339 m r x x 6 A October 7, 2024 5
Solution (Contd.) y 3 m y = x2 / 12 x 6 m Moment of inertia of the strip about its base is, 1 1 dIx= 3 3 ( ) (Fomula : ) dx y bd 3 3 6 6 6 6 7 1 1 1 1 x x dx = = = = = 3 4 7 714 . m I dI y dx x x 3 3 3 3 12 3 12 7 0 0 0 . 7 714 I = = = . 1 339 m r x x 6 A October 7, 2024 6
Problem-3 Calculate the moments of inertia of the area of a circle about a diametral axis and about the polar axis through the center. Specify the radii of gyration. 10/7/2024 7
Solution r 4 r 2 2 = = = 2 ( ) I r dA r r dr 0 0 0 0 z 2 0 Radius of gyration is 4 I r r = = = k z z 2 2 A r 2 = = + symmetry By and we know I I I I I x y z x y 4 4 1 1 r r = + = = = = 2 I I I I I I z x x x x z 2 2 2 4 Radius of gyration is 4 I r r = = = k x x 2 4 2 A r 10/7/2024 8
Moment of Inertia of Composite Areas Important Formulas October 7, 2024 9
Problem-4 Determine the moment of inertia of the shaded area about the x-axis. y 100 mm 20 mm 50 mm 20 mm 170 mm 20 mm x O October 7, 2024 10
Solution y 1 100 mm = = 3 6 4 20 mm 20 240 92 16 . 10 mm I 1 x 3 3 50 mm 2 1 = + = 3 2 6 4 60 20 ( 60 20 ) 230 63 52 . 10 mm I 20 mm 2 x 12 1 1 = + = 3 2 6 4 170 mm 20 70 ( 70 20 ) 205 59 41 . 10 mm I 3 x 12 20 mm = + + = 6 4 215 09 . 10 mm I I I I Ans. x 1 2 3 x x x x O October 7, 2024 11
