Question

10–53. Determine the moment of inertia of the area about axis. the y у 3 in.3 in.-- 6 in. 2 in. 4 in. Probs. 10-52/53

Answer 1

So lets label our figure and set it up as follows

10–53. Determine the moment of inertia of the area about the y axis. у 3 in.3 in.-- D B 6 in. 0 E 2 in. 4 in. X А F Probs. 10-52/53

We will calculate moment of intertia as follows

Momement of inertia of blue shaded area = Moment of intertia of big rectangle ABCG + Moment of inertia of Triangle COE + Moment of Inertia of rectangle GOEF- Moment of inertia of circle

So we have the areas as follows. Use -ve notation circle

Part Area Type Area formula Area (in) 1 2 3 3 Rectangle ABCG Triangle COE Rectangle GOEF Circle AB * BC 1/2 *CO * DE GO *OE pi * radius? 30 9 12 -12.571 38.429

We can then calculate the centroids as follows

Part Area Type Area formula Area (in?) Centroid x cooordiate Centroid y cooordiate 1 1.50 2 3 3 Rectangle ABCG Triangle COE Rectangle GOEF Circle AB * BC 1/2 *CO * DE GO *OE pi * radius? 30 9 12 -12.571 38.429 4.00 4.50 3.00 5.00 6.00 2.00 4.00

For rectangle ABCG I_yc = 1/12 * 10 * 6³ = 180

For triangle COE I_yc (of Triangle COE) = bh³/36 = 1/36* 3 * 6³ = 18

For rectangle GOEF I_yc (of rectangle GOEF) = 1/2 * 4 * 3³ = 54

For circle I_yc of circle = pi* radius⁴/4 = 22/7 * 2⁴/4 = 12.571

So we have the following

Part Area Type Area formula Area (in) Centroid x cooordiate Centroid y Area * Area * cooordiate Centroid(x) Centroidly) Moment of Inertia lyc 180 18 2 3 3 Rectangle ABCG Triangle COE Rectangle GOEF Circle AB * BC 1/2 *CO * DE GO *OE pi * radius 30 9 12 1.50 4.00 4.50 3.00 5.00 6.00 2.00 4.00 45.000 36.000 54.000 -37.714 97.286 150.000 54.000 24.000 -50.286 177.714 54 12.571 -12.571 38.429

by parallel axes theorem we know that

I_y = I_yc + (dx)² *Area

so we have

Part Area Type Area formula Area (in) Centroid x cooordiate Centroid y Area * Area * cooordiate Centroid(x) Centroidly) Moment of Inertia lyc dx Moment of Inertia Value 30 3 9 4 2 3 3 Rectangle ABCG Triangle COE Rectangle GOEF Circle AB * BC 1/2 *CO * DE GO *OE pi * radius 1.50 4.00 4.50 3.00 5.00 6.00 2.00 4.00 45.000 36.000 54.000 -37.714 97.286 150.000 54.000 24.000 -50.286 177.714 180 18 54 12.571 12 -12.571 38.429 4.5 450.000 162.000 297.000 -100.572 808.428 3

So moment of inertia of the shaded area = 800.428

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