Explanation Given information: The thin bent plate with uniform density, the weight of both the plates is W , the origin is O , the plate is square with dimension a unit, the centroidal axis for plate 1 is x ′ , y ′ , z ′ , the centroidal axis for plate 2 is x ″ , y ″ , z ″ , the perpendicular distance between the centroidal point P 1 and y -axis is 0 , the perpendicular distance between centroidal point P 2 and x -axis is 0 . The figure below illustrates different centroidal axis. Figure-(1) Write the expression for the mass of plate 1. m 1 = W 2 g ....... (I) Here, the weight of both the plates is W , the acceleration due to gravity is g , the mass of the plate 1 is m 1 . Write the expression for the mass of plate 2. m 2 = W 2 g ....... (II) Here, the weight of both the plates is W , the acceleration due to gravity is g , the mass of the plate 1 is m 2 . Write the expression of position vector of the line joining O A . O A → = x 1 i ^ + y 1 j ^ + z 1 k ^ ....... (III) Here, the vector along the line joining the points O and A is O A → , the unit vector along x direction is i ^ , the unit vector along y direction is j ^ , the unit vector along z direction is k ^ , the coordinate on x -axis is x 1 , the coordinate on y -axis is y 1 , the coordinate on z -axis is z 1 . Write the expression for the magnitude of position vector O A → . | O A → | = x 1 2 + y 1 2 + z 1 2 ....... (IV) Here, the magnitude of the position vector O A → is | O A | → . Write the expression for unit vector along O A → . η ^ OA = O A → | O A → | ....... (V) Here, the unit vector along O A is η ^ OA . Write the expression of mass moment of inertia of section 1 about x -axis. ( I x ) 1 = m 1 L 1 2 12 + m 1 ( L 1 2 ) 2 ....... (VI) Here, the mass moment of inertia of section 1 about x -axis is ( I x ) 1 , the dimension of square plate section 1 is L 1 . Write the expression of mass moment of inertia of section 2 about x -axis. ( I x ) 2 = m 2 12 ( L 2 2 + L 2 2 ) + m 2 ( ( L 2 2 ) 2 + ( L 2 2 ) 2 ) ....... (VII) Here, the mass moment of inertia of section 2 about x -axis is ( I x ) 2 , the dimension of square plate section 2 is L 2 . Write the expression of total mass moment of inertia about x -axis. I x = ( I x ) 1 + ( I x ) 2 ....... (VIII) Here, the total mass moment of inertia about x -axis is I x . Write the expression of mass moment of inertia of section 1 about y -axis. ( I y ) 1 = m 1 12 ( L 1 2 + L 1 2 ) + m 1 ( ( L 1 2 ) 2 + ( L 1 2 ) 2 ) ....... (IX) Here, the mass moment of inertia of section 1 about y -axis is ( I y ) 1 . Write the expression of mass moment of inertia of section 2 about y -axis. ( I y ) 2 = m 2 12 ( L 2 2 ) + m 2 ( L 2 2 + ( L 2 2 ) 2 ) ....... (X) Here, the mass moment of inertia of section 2 about y -axis is ( I y ) 2 . Write the expression of total mass moment of inertia about y -axis. I y = ( I y ) 1 + ( I y ) 2 ....... (XI) Here, the total mass moment of inertia about y -axis is I y . Write the expression of mass moment of inertia of section 1 about z -axis. ( I z ) 1 = m 1 L 1 2 12 + m 1 ( L 1 2 ) 2 ....... (XII) Here, the mass moment of inertia of section 1 about z -axis is ( I z ) 1 . Write the expression of mass moment of inertia of section 2 about z -axis. ( I z ) 2 = m 1 L 2 2 12 + m 1 ( L 2 + ( L 2 2 ) 2 ) ....... (XIII) Here, the mass moment of inertia of section 2 about z -axis is ( I z ) 2 . Write the expression of total mass moment of inertia about z -axis. I z = ( I z ) 1 + ( I z ) 2 ....... (XIV) Here, the total mass moment of inertia about z -axis is I z . From, the symmetry in above figure about x ′ , y ′ and z ′ . ( I x ′ y ′ ) 1 = ( I y ′ z ′ ) 1 = ( I z ′ x ′ ) 1 = 0 ....... (XV) Here, the product mass moment of inertia in x ′ - y ′ plane is ( I x ′ y ′ ) 1 , the product mass moment of inertia in y ′ - z ′ plane is ( I y ′ z ′ ) 1 , the product mass moment of inertia in z ′ - x ′ plane is ( I z ′ x ′ ) 1 . From, the symmetry in above figure about x ″ , y ″ and z ″ . ( I x ″ y ″ ) 2 = ( I y ″ z ″ ) 2 = ( I z ″ x ″ ) 2 = 0 ....... (XVI) Write the expression for product of mass moment of inertia in x - y plane. ( I xy ) = ∑ i = 1 i = 2 ( ( I x ′ y ′ ) i + m i x i ¯ y i ¯ ) ( I xy ) = ( ( I x ′ y ′ ) 1 + m 1 x 1 ¯ y 1 ¯ ) + ( ( I x ′ y ′ ) 2 + m 2 x 2 ¯ y 2 ¯ ) ....... (XVII) Here, the product mass moment of inertia is ( I xy ) , the subscript limit is i . Write the expression for product mass moment of inertia in y - z plane. ( I yz ) = ∑ i = 1 i = 2 ( ( I y ′ z' ) i + m i y i ¯ z i ¯ ) ( I yz ) = ( ( I y ′ z ′ ) 1 + m 1 y 1 ¯ z 1 ¯ ) + ( ( I y ′ z ′ ) 2 + m 2 y 2 ¯ z 2 ¯ ) ....... (XVIII) Here, the product mass moment of inertia is ( I yz ) , the subscript limit is i . Write the expression for product mass moment of inertia in z - x plane. ( I zx ) = ∑ i = 1 i = 2 ( ( I z' x ′ ) i + m i x i ¯ z i ¯ ) ( I zx ) = ( ( I z ′ x ′ ) 1 + m 1 x 1 ¯ z 1 ¯ ) + ( ( I z ′ x ′ ) 2 + m 2 x 2 ¯ z 2 ¯ ) ....... (XIX) Here, the product mass moment of inertia in z - x plane is ( I zx ) . Write the expression of mass moment of inertia about O A . I OA = I x λ x 2 + I y λ y 2 + I z λ z 2 − 2 I xy λ x λ y − 2 I yz λ z λ y − 2 I xz λ x λ z ....... (XX) Here, the mass moment of inertia about O A is I OA , the coefficient of i ^ in unit vector η O A is λ x , the coefficient of j ^ in unit vector η O A is λ y , the coefficient of k ^ in unit vector η O A is λ z . Calculation: Substitute a for x 1 , a for y 1 , a for z 1 in Equation (III). O A → = a i ^ + a j ^ + a k ^ Substitute a for x 1 , a for y 1 , a for z 1 in Equation (IV). | O A → | = a 2 + a 2 + a 2 = 3 a 2 = 3 a Substitute a i ^ + a j ^ + a k ^ for O A → , 3 a for | O A → | in Equation (V). η ^ OA = a i ^ + a j ^ + a k ^ 3 a = a ( i ^ + j ^ + k ^ ) 3 a = 1 3 i ^ + 1 3 j ^ + 1 3 k ^ Substitute a for L 1 in Equation (VI)

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ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

Publisher: Mcgraw-hill Education,

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Chapter B, Problem B.57P

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The mass moment of inertia with respect to the line joining the origin O and point A.

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