Explanation Given: The height of the shaded area is a . The width of the shaded area is a . Show the area of the differential element that is parallel to the y -axis as in Figure (1). From Figure 1, Compute the area of the differential element that parallel to the y -axis as shown with shaded area from Figure 1. d A = y d x (I) Here, the area of the differential element is d A . Substitute ( a 1 2 − x 1 2 ) 2 for y in Equation (I). d A = ( a 1 2 − x 1 2 ) 2 d x Compute the coordinate of the centroid of the differential element along the x axis. x ˜ = x Compute the coordinate of the centroid of the differential element along the y axis. y ˜ = y 2 (II) Substitute ( a 1 2 − x 1 2 ) 2 for y in Equation (II). y ˜ = ( a 1 2 − x 1 2 ) 2 2 = 1 2 ( a 1 2 − x 1 2 ) 2 Compute the product of inertia of differential element with respect to the x and y axes. d I x y = d I ¯ x ′ y ′ + d A x ˜ y ˜ (III) Here, the first integral of the moment of inertia of the area about the centroidal x and y axes is I ¯ x ′ y ′ . Substitute 0 for d I ¯ x ′ y ′ , ( a 1 2 − x 1 2 ) 2 d x for d A , x for x ˜ , and 1 2 ( a 1 2 − x 1 2 ) 2 for y ˜ in Equation (III). d I x y = 0 + [ ( a 1 2 − x 1 2 ) 2 d x ] ( x ) [ 1 2 ( a 1 2 − x 1 2 ) 2 ] = [ ( a + x − 2 a 1 2 x 1 2 ) d x ] x [ 1 2 ( a + x − 2 a 1 2 x 1 2 ) ] = 1 2 ( x 3 + a 2 x + 6 a x 2 − 4 a 3 2 x 3 2 − 4 a 1 2 x 5 2 ) d x Conclusion: Compute the product of inertia of the shaded area with respect to x and y axes. I x y = ∫ 0 a d I x y (IV) Substitute 1 2 ( x 3 + a 2 x + 6 a x 2 − 4 a 3 2 x 3 2 − 4 a 1 2 x 5 2 ) d x for d I x y in Equation (IV)

14th Edition

ISBN: 9780135681879

Author: HIBBELER

Publisher: PEARSON

See similar textbooks

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…

Videos

Question

Chapter 10.7, Problem 59P

To determine

The product of inertia of the shaded area with respect to x and y axes.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.7, Problem 58P

Chapter 10.7, Problem 60P

Students have asked these similar questions

ing Design 8-7 Problem 8-7 485 / 1109 157% A screw clamp similar to the one shown in the figure has a handle with diameter in made of cold-drawn AISI 1006 steel. The overall length is 4.25 in. The screw is in-10 UNC and is 8 in long, overall. Distance A is 3 in. The clamp will accommodate parts up to 6 in high. (a) What screw torque will cause the handle to bend permanently? (b) What clamping force will the answer to part (a) cause if the collar friction is neglected and if the thread friction is 0.15? (c) What clamping force will cause the screw to buckle? (d) Are there any other stresses or possible failures to be checked? ट Activate Windows Go to Settings to activat Windows. ENG 8:58 PM

A billet 75mm long and 25mm in diameter is to be extruded in a direct extrusion operation with extrusion ratio re= 4.0 .The extrudate has a round cross section, the die angle (half angle )is 90 degree .The work metal has a strength coefficient of 415 Mpa, and strain hardening exponent of 0.18. Use the Johnson formular with a= 0.8 and b=1.5 to estimate extrusion strain. Find the Pressuer applied to the end of the billet as the ram moves forward.

14- effect of different carburizing treatments on the microstructure and mechanical properties of a case produced by carburizing. 5-surface film between two surfaces is one of the main factors affecting wear.

Chapter 10 Solutions

ENGINEERING MECHANICS: STATICS

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

The product of inertia of the shaded area with respect to x and y axes.

Solution Summary: The author explains the product of inertia of the shaded area with respect to the x and y axes.

Concept explainers

Videos

The product of inertia of the shaded area with respect to x and y axes.

Want to see the full answer?

Chapter 10 Solutions