The curve \( y = x^4 \) bounds the shaded section on top. The vertical lines \( x = 2 \, \text{in.} \) and \( x = 5 \, \text{in.} \) bound the shaded section on the left and right respectively. The x-axis bounds the shaded section below. For the shaded section determine the moment of inertia with respect to the x-axis, \( I_x \); the moment of inertia with respect to the y-axis, \( I_y \); the polar moment of inertia, \( J_o \); and the radius of gyration with respect to the x-axis, \( k_x \). Show all work.### Diagram Explanation:The graph displayed shows the curve \( y = x^4 \), plotted on a coordinate system with the x-axis labeled \( x \, (\text{in.}) \) and the y-axis labeled \( y \, (\text{in.}) \). The shaded area lies between the curve and the x-axis, starting from \( x = 2 \, \text{in.} \) and ending at \( x = 5 \, \text{in.} \). The shape tapers as it follows the function from left to right, with increasing values as \( x \) increases, ending in a sharp corner at \( x = 5 \, \text{in.} \).

Given Data: The equation of the curve is y=x4. The limit of curve in X-direction is 2 in. to 5 in. The limit of curve in Y-direction is 0 to x4. Calculate the Moment of Inertia for the shaded area. The moment of inertia with respect to X-axis is IX IX=∬y2dxdy=∫25∫0x4y2dydx=∫25y33dx=∫25x433dx=13∫25x12dx=13x1313=13513-21313=13×122069493313=3.13×107 in4 The moment of inertia with respect to Y-axis is IY IY=∬x2dxdy=∫25∫0x4x2dydx=∫25x2ydx=∫25x2×x4dx=∫25x6dx=x77=57-277=779977=1.114×104 in4 The polar moment of inertia is J0 J0=IX+IY=3.13×107+1.114×104=3.131×107 in4Calculate the radius of gyration with respect to X-axis. The area of the shaded portion is A A=∬dxdy=∫25∫0x4dydx=∫25ydx=∫25x4dx=x55=55-255=30935=618.6 in2 The radius of gyration with respect to X-axis is kx kx=IXA=3.13×107 in4618.6 in2=50598.125=224.94 in

Engineering

Civil Engineering

The curve y= x* bounds the shaded 600 section on top. The vertical lines X = 2 in. and x = 5 in. bound the shaded section on the left and right respectively. 500 - y = x* The x-axis bounds the shaded section 400 - below. For the shaded section 300 - determine the moment of inertia with respect to the X-axis, I, ; the moment of inertia with respect to the y-axis, I,; the polar moment of inertia, J,; and the radius of gyration with respect to the x- axis, k̟. Show all work. 200 - 100 1 2 6 x (in.) y (in.)

The curve y= x* bounds the shaded 600 section on top. The vertical lines X = 2 in. and x = 5 in. bound the shaded section on the left and right respectively. 500 - y = x* The x-axis bounds the shaded section 400 - below. For the shaded section 300 - determine the moment of inertia with respect to the X-axis, I, ; the moment of inertia with respect to the y-axis, I,; the polar moment of inertia, J,; and the radius of gyration with respect to the x- axis, k̟. Show all work. 200 - 100 1 2 6 x (in.) y (in.)

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Concept explainers

Mechanics of Solids

It is the study of the behavior of solid materials, their characteristics under various circumstances such as variation in external forces, temperature, and other internal factors. It is one of most important study in the field of engineering science, especially chemistry and mechanical engineering. In rigid body, the idealization of the solid body in which deformation is neglected. Deformation is known as the force acting on the solid state system which exerts change in its shape of the body but it continues to be at rest or in a uniform motion but it cannot accelerate. But those solid systems do have elasticity, compressibility and can be bent. This field is very vast since there are wide range of materials around us.

Question

The curve \( y = x^4 \) bounds the shaded section on top. The vertical lines \( x = 2 \, \text{in.} \) and \( x = 5 \, \text{in.} \) bound the shaded section on the left and right respectively. The x-axis bounds the shaded section below. For the shaded section determine the moment of inertia with respect to the x-axis, \( I_x \); the moment of inertia with respect to the y-axis, \( I_y \); the polar moment of inertia, \( J_o \); and the radius of gyration with respect to the x-axis, \( k_x \). Show all work.

### Diagram Explanation:

The graph displayed shows the curve \( y = x^4 \), plotted on a coordinate system with the x-axis labeled \( x \, (\text{in.}) \) and the y-axis labeled \( y \, (\text{in.}) \). The shaded area lies between the curve and the x-axis, starting from \( x = 2 \, \text{in.} \) and ending at \( x = 5 \, \text{in.} \). The shape tapers as it follows the function from left to right, with increasing values as \( x \) increases, ending in a sharp corner at \( x = 5 \, \text{in.} \).

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