(a) Explanation Given information: The height of the section is b . The half width of the section is a . Calculation: Express the area ( A ) of the semielliptical plate as below: A = π a b 2 Express the mass ( m ) as below: m = ρ t A Here, ρ is density, t thickness of the semielliptical plate and A is area of the semielliptical plate. Rewrite the above equation as, ρ t = m A Substitute π a b 2 for A . ρ t = m π a b 2 = 2 m π a b Express the mass moment of inertia ( I mass ) as below: I mass = ρ t I a r e a Substitute 2 m π a b for ρ t . I mass = 2 m π a b I a r e a (1) Express the centroid of semielliptical plate ( y ¯ ) about xx axis as below: y ¯ = 4 b 3 π Express the moment of inertia of area ( I x , a r e a ) about xx axis as below: I x , a r e a = π 8 a b 3 Express the moment of inertia ( I x , area ) of area of semi elliptical plate with respect to x axis as below: I x , area = I B B ' , area + A y ¯ 2 . (2) Calculate the expression for the moment of inertia ( I B B ' , area ) of area of semi elliptical plate with respect to B B ' axis as below: Rewrite the equation (2) as , I B B ' , area = I x , area − A y ¯ 2 Substitute π 8 a b 3 for I x , a r e a , π a b 2 for A and 4 b 3 π for y ¯ (b) To determine Find the mass moment of inertia ( I B B ' , mass ) of the plate with respect to the centroidal axis C C ' that is perpendicular to the plate.

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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Chapter 9.5, Problem 9.113P

(a)

To determine

The mass moment of inertia (IBB',mass) of the plate with respect to the centroidal axis BB'

(b)

To determine

Find the mass moment of inertia (IBB',mass) of the plate with respect to the centroidal axis CC' that is perpendicular to the plate.

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Chapter 9.5, Problem 9.112P

Chapter 9.5, Problem 9.114P

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The mass moment of inertia ( I B B ' , mass ) of the plate with respect to the centroidal axis B B '