Consider the shaded area shown in Figure 1 Suppose that a 9

Consider the shaded area shown in (Figure 1). Suppose that a = 9 in. b = 4 and r = 3 in. Part A Determine the moment of inertia for the shaded area about the y axis.

Solution

First, we divide the composite shape into simpler shapes.

1. A right triangle with sides a and b+b+b = 3b

2. A circle of radius r

3. A square of sides b+b = 2b

4. A right triangle of sides b and b+b = 2b

Now we find Iy for each of these shapes.

1.

For the right triangle with sides a and 3b, Iy about the 3b side is given by

Iy = a³ * (3b) / 12 = (a³b /4)

2.

For the circle, Inertia about its dia = pi*r⁴ /4

Area of circle = pi*r²

Distance of centre of circle to y-axis = b

By parallel axis theorem, for circle Iy = (pi*r⁴ /4) + (pi*r²)*b² = (pi*r²)*(b² + r² /4)

3.

For the square of sides 2b, Iy = (2b)(2b)³ /3 = (16/3)b⁴

4.

For the right triangle with sides b and 2b, we get Iy = b*(2b)³ /12 = (2/3)b⁴

Now putting it all together, we get

Iy = (a³b /4) - (pi*r²)*(b² + r² /4) + (16/3)b⁴ + (2/3)b⁴

Putting values, we get Iy = 1748.993 in⁴