Concept explainers
Show that the sum of the moments of inertia of a body, Ixx + Iyy + Izz, is independent of the orientation of the x, y, z axes and thus depends only on the location of the origin.
Show that the sum of moments of inertia of a body,
Answer to Problem 1P
The given equation is proved.
Explanation of Solution
Given:
The moment of inertia of a body with respect to
The moment of inertia of a body with respect to
The moment of inertia of a body with respect to
Conclusion:
Express the summation of moments of inertia of body with respect to
Here, the distance from the origin to
Since the distance
Hence, the given expression is satisfied and proved.
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Chapter 21 Solutions
Engineering Mechanics Dynamics
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- Find the moments of inertia Ix, Iy, I0 for a lamina in the shape of an isosceles right triangle with equal sides of length a if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse. (Assume that the coefficient of proportionality is k, and that the lamina lies in the region bounded by x = 0, y = 0, and y = a − x).Ix = Iy = I0 =arrow_forwardHow do you determine the moment of inertia of a plane surface?arrow_forwardUsing the polar moment of inertia of the isosceles triangle of Prob. 9.28, show that the centroidal polar moment of inertia of a circular area of radius r is π4/2. ( Hint: As a circular area is divided into an increasing number of equal circular sectors, what is the approximate shape of each circular sector?)(Reference to Problem 9.28):Determine the polar moment of inertia and the polar radius of gyration of the isosceles triangle shown with respect to point O.arrow_forward
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