**Exercises:****10–30.** Determine the moment of inertia for the beam’s cross-sectional area about the x axis.**10–31.** Determine the moment of inertia for the beam’s cross-sectional area about the y axis.**Diagram Explanation:**The diagram shows a cross-sectional area of a beam with an L-shaped geometry. The beam consists of two rectangular sections that form an L shape, oriented such that one leg is vertical and the other is horizontal.- The horizontal section is 10 inches long and 1 inch thick.- The vertical section is 8 inches tall and 1 inch thick.- The overall height from the bottom to the top of the structure is 8 inches for the longer vertical section.- There is an additional vertical part aligned with the horizontal base, which is 3 inches in height and 1 inch thick.Axes are labeled:- The x-axis runs horizontally along the base.- The y-axis runs vertically, intersecting the L-shape at a corner where the 3-inch vertical part begins.The task is to calculate the moment of inertia of this cross-sectional area about the x and y axes, respectively, for exercises 10-30 and 10-31.

It is required to determine moment of inertia about x and y axisTo determine the moment of inertia of cross-sectionwe divide it into three parts1. Rectangle having dimension (B×1)2. Rectangle having dimension (1×8)3. Rectangle having dimension (3×1)Moment of inertia about x axis for each segment can be find out using parallel axis theoremIx=IX1 +Ady2Segment Ai (in2) (dy)i (IX1 )i(in4) (Ady2)i in4 (Ix )i in41 1(8) 4 112 (1)83 128 170.672 8(1) 0.5 112 (8)13 2 2.67 3 1(3) 1.5 112 (1)33 6.75 9.0So, Ix =Σ(Ix)i = 170.67+2.67+9 = 182.33 in4Moment of inertia about y axis for each segment can be determined using parallel axis theoremIy =Iy1 +Adx2Segment Ai (in2) (dx)i in (Iy1 )i in4 (Adx2 )i in4 (Iy)i in4 1 8(1) 9.5 112 (8)(1)3 722 722.672 1(8) 5 112 (1)(8)3 200 242.67 3 3(1) 0.5 112 (3)(1)3 0.75 1So, Iy =Σ(Iy)i = 722.67+242.67 = 966.33 in4

Engineering

Mechanical Engineering

10-30. Determine the moment of inertia for the beam's cross-sectional area about the x axis. 10-31. Determine the moment of inertia for the beam's cross-sectional area about the y axis. coubnic blotnas y 1 in. -1 in. 8 in. 3 in. 10 in. 1 in.

10-30. Determine the moment of inertia for the beam's cross-sectional area about the x axis. 10-31. Determine the moment of inertia for the beam's cross-sectional area about the y axis. coubnic blotnas y 1 in. -1 in. 8 in. 3 in. 10 in. 1 in.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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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 provide resistance in changing its state of velocity or motion for coming to a state of rest. In other words, Newton's first law of motion describes the law of inertia of a rigid body.

Question

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**Exercises:**

**10–30.** Determine the moment of inertia for the beam’s cross-sectional area about the x axis.

**10–31.** Determine the moment of inertia for the beam’s cross-sectional area about the y axis.

**Diagram Explanation:**

The diagram shows a cross-sectional area of a beam with an L-shaped geometry. The beam consists of two rectangular sections that form an L shape, oriented such that one leg is vertical and the other is horizontal.

- The horizontal section is 10 inches long and 1 inch thick.
- The vertical section is 8 inches tall and 1 inch thick.
- The overall height from the bottom to the top of the structure is 8 inches for the longer vertical section.
- There is an additional vertical part aligned with the horizontal base, which is 3 inches in height and 1 inch thick.

Axes are labeled:
- The x-axis runs horizontally along the base.
- The y-axis runs vertically, intersecting the L-shape at a corner where the 3-inch vertical part begins.

The task is to calculate the moment of inertia of this cross-sectional area about the x and y axes, respectively, for exercises 10-30 and 10-31.

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