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Engineering Mechanics: Statics Assignment 6 Chapter 6
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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.)
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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.
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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
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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:
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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:
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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: