Explanation Given information: The curve equation is y = k ( x − a ) 3 . Calculation: Sketch the vertical strip shaded portion as shown in Figure 1. Refer to Figure 1. Consider a differential element to y axis width x and thickness dx as shown in Figure. Refer to Figure 1. At x = 3 a , y = b . Find the constant k using the relation: y = k ( x − a ) 3 (1) Substitute b for y and 3 a for x in Equation (1). b = k ( 3 a − a ) 3 b = k ( 2 a ) 3 b = 8 a 3 k = b 8 a 3 Substitute b 8 a 3 for k in Equation (1). y = b 8 a 3 ( x − a ) 3 Determine the area of the strip element d A as shown in below: d A = y d x Determine the moment of inertia ( d I y ) with respect to y axis of a rectangular strip: d I y = x 2 d A (2) Substitute y d x for d A in Equation (2). d I y = x 2 y d x (3) Substitute b 8 a 3 ( x − a ) 3 for y in Equation (3). d I y = x 2 ( b 8 a 3 ( x − a ) 3 ) d x = b 8 a 3 x 2 ( x 3 − 3 x 2 a + 3 x a 2 − a 3 ) d x Integrate both sides of the equation (3) with respect to x . I y = d I y = ∫ a 3 a b 8 a 3 x 2 ( x 3 − 3 x 2 a + 3 x a 2 − a 3 ) d x = ∫ a 3 a b 8 a 3 ( x 5 − 3 x 4 a + 3 x 3

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Moment of Inertia

Inertia can be termed as a resistance offered by a body to change its state. A rigid body initially at rest will apply resistance when acted by an external force that tends to change its state of rest or velocity, or a body initially at motion will provi…

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Chapter 9.1, Problem 9.12P

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Find the moment of inertia of the shaded area with respect to y axis by using direct integration.

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Chapter 9.1, Problem 9.11P

Chapter 9.1, Problem 9.13P

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