Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Concept explainers

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…
Question
### Problem Statement: 1. For the isosceles triangle shown to the right, find the moments of inertia, \( I_x \) and \( I_y \), about the centroidal axes. Hint: break into two right triangles and use the parallel axis theorem. **Answer:** \[ I_x = \frac{bh^3}{36}, \quad I_y = \frac{hb^3}{48} \] ### Diagram Explanation: The image features an isosceles triangle oriented in the Cartesian coordinate plane, with its base parallel to the x-axis and its vertex pointing upwards along the y-axis. The centroid of the triangle is labeled as point \( C \). #### Key Features: - **Axes:** - The y-axis is vertical, and the x-axis is horizontal, intersecting at the centroid point \( C \). - **Dimensions:** - The triangle's base is divided into two equal segments, each labeled \( \frac{b}{2} \). - The height of the triangle is divided into parts; from the base to the centroid is \( \frac{h}{3} \), and from the centroid to the vertex is \( \frac{2h}{3} \). - **Geometry Insight:** - To find the moments of inertia \( I_x \) and \( I_y \), the problem suggests breaking the triangle into two right triangles and applying the parallel axis theorem. This approach utilizes symmetry and centroid location to simplify the calculation of the moments of inertia for the isosceles triangle.
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Transcribed Image Text:### Problem Statement: 1. For the isosceles triangle shown to the right, find the moments of inertia, \( I_x \) and \( I_y \), about the centroidal axes. Hint: break into two right triangles and use the parallel axis theorem. **Answer:** \[ I_x = \frac{bh^3}{36}, \quad I_y = \frac{hb^3}{48} \] ### Diagram Explanation: The image features an isosceles triangle oriented in the Cartesian coordinate plane, with its base parallel to the x-axis and its vertex pointing upwards along the y-axis. The centroid of the triangle is labeled as point \( C \). #### Key Features: - **Axes:** - The y-axis is vertical, and the x-axis is horizontal, intersecting at the centroid point \( C \). - **Dimensions:** - The triangle's base is divided into two equal segments, each labeled \( \frac{b}{2} \). - The height of the triangle is divided into parts; from the base to the centroid is \( \frac{h}{3} \), and from the centroid to the vertex is \( \frac{2h}{3} \). - **Geometry Insight:** - To find the moments of inertia \( I_x \) and \( I_y \), the problem suggests breaking the triangle into two right triangles and applying the parallel axis theorem. This approach utilizes symmetry and centroid location to simplify the calculation of the moments of inertia for the isosceles triangle.
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