worksheet 5 - wordpress.com mechanics: statics assignment 6 chapter 6 page 1 of 12 worksheet 5 on...

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 1 of 12

Worksheet 5

On Chapter 6: Area Moments of Inertia

Part I: Theoretical Questions

1. The radius of gyration, say, about the x-axis, Kx is the value of an average distance from

the x axis such that ∫y2dA = Kx

2A. However is it possible for the value Kx

2A to be greater

than ∫y2dA? If yes explain why, if no explain why not.

2. What is the parallel axis theorem? State the parallel axis theorem for Ix, Iy, and Ixy.

3. In the angle section below the x-y axes, the x′-y′ axes, and the a-b axes are all

rectangular, and have the same origin, O as shown. θ here is an arbitrary rotating angle

while θm is the rotation angle required to get to the principal axes of inertia about the

origin O. Draw a Mohr’s circle showing the x-y axes, the x′-y′ axes, the a-b axes, θ, and

θm. (Remark: This will help you understand the physical significance of Mohr’s circle.)

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 2 of 12

Part II: Numerical Analysis

Note: Here, vector quantities are represented in bold-face letters whereas their scalar

counterparts are represented by normal-face letters.

1. Show how the equations for the principal moments of inertia are derived. Note that I have

provided here the detailed solution. You are expected to study it and rewrite it using your

own understanding.

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 3 of 12

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 4 of 12

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 5 of 12

2. Calculate the moment of inertia about the x-axis of the area enclosed between the y-axis

and the circular arcs of radius a whose centers are at O and A. Note that I have provided

here the detailed solution. You are expected to study it and rewrite it using your own

understanding.

Answer: Ix = 0.0969a4

a

a/2 a

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 6 of 12

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 7 of 12

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 8 of 12

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 9 of 12

3. Determine the moments of inertia of the shaded area about the y- and y′-axes. Note that I

have provided here the detailed solution. You are expected to study it and rewrite it using

your own understanding.

Answer:

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 10 of 12

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 11 of 12

4. Determine the product of inertia of the trapezoidal area about the x-y axes. Note that I

have provided here the detailed solution. You are expected to study it and rewrite it using

your own understanding.

Answer:

Engineering Mechanics: Statics Assignment 6 Chapter 6

Page 12 of 12

5. Determine the maximum moment of inertia about an axis through O and the angle α to

this axis for the triangular area shown. (Recommendation: Use Mohr’s circle)

Answer: