Explanation Show the centroidal location of the given section as Figure 1. Consider the section 1. Calculate the moment of inertia of the section 1 ( I x ) 1 using the formula: ( I x ) 1 = b 1 d 1 3 12 = 1 12 ( 2 a ) ( 2 a ) 3 = 2 a ( 8 a 3 ) 12 = 4 3 a 4 The expression for the moment of inertia of the section AA ( I A A ) 2 and section BB ( I B B ) 3 as follows; The section AA and BB compressed of semi-circle. ( I A A ) 2 = ( I B B ) 3 = π 8 a 4 The expression for the area of the semi-circle ( A ) as follows; A = π 2 a 2 The expression for the centroid of the semi-circle ( d ) as follows; d = 4 a 3 π Calculate the moment of inertia of the section 2 ( I ¯ x 2 ) 2 and section 3 ( I ¯ x 3 ) 3 using the formula: ( I A A ) 2 = ( I ¯ x 2 ) 2 + A d 2 ( I ¯ x 3 ) 3 = ( I ¯ x 2 ) 2 = ( I A A ) 2 − A d 2 Substitute π 8 a 4 for ( I A A ) 2 , π 2 a 2 for A and 4 a 3 π for d . ( I ¯ x 3 ) 3 = ( I ¯ x 2 ) 2 = π 8 a 4 − π 2 a 2 ( 4 a 3 π ) 2 = π 8 a 4 − π a 2 2 ( 16 a 2 9 π 2 ) = π 8 a 4 − 8 a 4 9 π = a 4 ( π 8 − 8 9 π ) Consider the section 2. The expression for the centroid of the section 2 ( d ) from neutral axis as follows; d = a + 4 a 3 π Calculate the moment of inertia of the section 2 ( I x ) 2 using the formula: ( I x ) 2 = ( I ¯ x 2 ) 2 + A d 2 Substitute a 4 ( π 8 − 8 9 π ) for ( I ¯ x 2 ) 2 , π 2 a 2 for A and ( a + 4 a 3 π ) for d . ( I x ) 2 = a 4 ( π 8 − 8 9 π ) + [ π 2 a 2 ( a + 4 a 3 π ) 2 ] = a 4 ( π 8 − 8 9 π ) + π 2 a 2 ( a 2 + ( 4 a 3 π ) 2 + 2 a 4 a 3 π ) = a 4 ( π 8 − 8 9 π ) + π 2 a 2 ( a 2 + 16 a 2 9 π 2 + 8 a 2 3 π ) = a 4 ( 4 3 + 5 π 8 ) Consider the section 3. The expression for the centroid of the section 3 ( d ) from neutral axis as follows; d = a − 4 a 3 π Calculate the moment of inertia of the section 3 ( I x ) 3 using the formula: ( I x ) 3 = ( I ¯ x 3 ) 3 + A d 2 Substitute a 4 ( π 8 − 8 9 π ) for ( I ¯ x 3 ) 3 , π 2 a 2 for A and ( a − 4 a 3 π ) for d

Vector Mechanics for Engineers: Statics and Dynamics

11th Edition

ISBN: 9780073398242

Author: Ferdinand P. Beer, E. Russell Johnston Jr., David Mazurek, Phillip J. Cornwell, Brian Self

Publisher: McGraw-Hill Education

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 9.2, Problem 9.35P

To determine

Find the moment of inertia of the shaded area with respect to x axis (Ix) and y axis (Iy).

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.2, Problem 9.34P

Chapter 9.2, Problem 9.36P

Students have asked these similar questions

6.76 A wind turbine is operating in a 12 m/s wind that has a den- sity of 1.2 kg/m³. The diameter of the turbine silhouette is 4 m. The constant-pressure (atmospheric) streamline has a diameter of 3 m upstream of the windmill and 4.5 m downstream. Assume that the velocity distributions are uniform and the air is incom- pressible. Determine the force on the wind turbine. m P = Patm 4 Vz 4m 4 m Fx. Problem 6.76

For the position shown in the figure the spring is unstretched. The spring constant k, is designed such that after the system is released from rest, the speed of the mass is zero just as the 0.6 slug mass touches the floor. Find the spring constant, k and the maximum speed of block A and the location (distance above floor) where this occurs.

||! Sign in MMB241 - Tutorial L9.pd X PDF MMB241 - Tutorial L10.pX DE MMB241 - Tutorial L11.p x PDF Lecture W12 - Work and X File C:/Users/KHULEKANI/Desktop/mmb241/MMB241%20-%20Tutorial%20L11.pdf PDE Lecture W11 - Power and X Draw Alla | Ask Copilot ++ 3 of 3 | D 6. If the 50-kg load A is hoisted by motor M so that the load has a constant velocity of 1.5 m/s, determine the power input to the motor, which operates at an efficiency € = 0.8. 1.5 m/s 2 7. The sports car has a mass of 2.3 Mg, and while it is traveling at 28 m/s the driver causes it to accelerate at 5m/s². If the drag resistance on the car due to the wind is FD= 0.3v²N, where v is the velocity in m/s, determine the power supplied to the engine at this instant. The engine has a running efficiency of P = 0.68. 8. If the jet on the dragster supplies a constant thrust of T-20 kN, determine the power generated by the jet as a function of time. Neglect drag and rolling resistance, and the loss of fuel. The dragster has a mass of 1…

Chapter 9 Solutions

Vector Mechanics for Engineers: Statics and Dynamics

Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - 9.1 through 9.4 Determine by direct integration...Ch. 9.1 - Prob. 9.3P Ch. 9.1 - Prob. 9.4P Ch. 9.1 - 9.5 through 9.8 Determine by direct integration...Ch. 9.1 - 9.5 through 9.8 Determine by direct integration...Ch. 9.1 - Prob. 9.7P Ch. 9.1 - Prob. 9.8P Ch. 9.1 - 9.9 through 9.11 Determine by direct integration...Ch. 9.1 - Prob. 9.10P

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.

Find the moment of inertia of the shaded area with respect to x axis ( I x ) and y axis ( I y ) .

Concept explainers

Videos

Find the moment of inertia of the shaded area with respect to x axis (Ix) and y axis (Iy).

Want to see the full answer?

Chapter 9 Solutions