Solved Example & Practice Problems: Mass Moment of Inertia of Composite Bodies

Mass Moment of Inertia of Composite Bodies

Solved Examples for Understanding

Example 7.5.1 

Find the mass moment of inertia of the rectangular block shown in Fig. 7.5.1 about the vertical Y axis. A cuboid of 20 × 20 × 20 mm has been removed from the block as shown in the Fig. 7.5.1. The mass density of the material of the block is 7850 kg/m3.

Solution: 

The centroidal Y-axes of the rectangular block and cuboid are shown in Fig. 7.5.1 (a).

For rectangular block,

m1 = 7850 × (0.1× 0.06 × 0.02) = 0.942 kg

Distance between Y-axis and y1 axis,

Moment of inertia about Y-axis is,

For cuboid,

m2 = 7850 × (0.02 × 0.02 × 0.02) = 0.0628 kg

Distance between Y-axis and y2 axis is

Example 7.5.2 

A steel forging consists of a 60 × 20 × 20 mm rectangular prism and two cylinders of diameter 20 mm and length 30 mm as shown in Fig. 7.5.2. Determine the moments of inertia of the forging with respect to the co-ordinate axes, knowing that the density of steel is 7850 kg/m3.

Solution: 

For rectangular prism, the mass is,

m1 = 7850 × (0.06 × 0.02 × 0.02) = 0.1884 kg

For cylinders A and B,

m2 = m3 = 7850 × (л × 0.012 × 0.03) = 0.074 kg

Moment of Inertia with respect to X-axis :

Moment of inertia with respect to Y-axis :

For prism,

Moment of Inertia with respect to Z-axis :

For prism,

Example 7.5.3

 A rectangular R.C.C. column is centrally cast over a concrete bed. R.C.C. in Fig. 7.5.3 column is of section 30 × 45 m and height 4 m. The concrete bed is of size 3 × 4.5 m and thickness 30 cm. Find the mass moment of inertia of the column and bed combination abuot its vertical centroidal axis. Mass density of concrete - 2500 kg/m3.

Solution: 

The vertical centroidal axis of both column and bed coincide with overall vertical centroidal axis due to symmetry.

For column

Volume = 0.3 × 0.45 × 4 = 0.54 m3

Mass = Volume × Density = 0.54 × 2500

= 1350 kg

Examples for Practice

Q.1 

Find the moment of inertia of a thin disc of mass 10 kg and radius 100 mm about. an axis passing through any point on its circumference and perpendicular to the plane of the disc. 

[Ans. : 0.15 kg-m2]

Q.2

Determine the moment of inertia of the compound pendulum shown in Fig. 7.5.4 about an axis through O, perpendicular to the plane of the paper. Mass of rod is 400 g and mass of disc is 200 g.

[Ans. : 0.0134 kg-m2]