Explanation Calculation: Sketch the element as shown in Figure 1. Find the differential element of mass ( d m ) as shown below: m = ρ V (1) Here, ρ is density and V is volume of cylinder. Differentiate with respect to V . d m = ρ d V (2) Substitute π a 2 d x for d V in Equation (2). d m = ( π a 2 d x ) ρ = ρ π a 2 d x Find the strip element ( d I y ) about y axis as shown below: d I y = d I ¯ y + x 2 d m (3) Substitute ( 1 4 a 2 d m ) for d I ¯ y in Equation (3). d I y = ( 1 4 a 2 d m ) + x 2 d m = ( 1 4 a 2 + x 2 ) ρ π a 2 d x (4) Integrate the Equation (4) and apply the limits. d I y = ∫ d I y = ∫ − L 2 L 2 ρ π a 2 ( 1 4 a 2 + x 2 ) d x = ρ π a 2 [ 1 4 a 2 x + x 3 3 ] − L 2 L 2 = 2 ρ π a 2 [ 1 4 a 2 x + x 3 3 ] 0 L 2

12th Edition

ISBN: 9781259977121

Author: BEER

Publisher: MCG

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 9.5, Problem 9.119P

To determine

Find the mass moment of inertia with respect to y axis of the right circular cylinder by using direct integration.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 9.5, Problem 9.118P

Chapter 9.5, Problem 9.120P

Students have asked these similar questions

Draw a counter balance circuit of a vertical cylinder. using counter balance valve and external load.

please sketch a stress-strain diagram for a typical structural steel in tension and display all of the important features.

Problem 1 (30 Points) Consider the following 2 scenarios. In scenario 1, a mass m slides on a cylindrical surface of radius R. In scenario 2, a mass m hangs at the end of a thin massless rod of length R. In both scenarios, there is no friction either on the surface (scenario 1), or at the pivot point of the pendulum (scenario 2). Also in both scenarios, there is one generalized coordinate, . R Scenario 1 R m R g Scenario 2 m HINT: In both scenarios, it is much easier to choose your datum for potential energy as the center of the bowl (scenario 1), or the pivot point of the pendulum (scenario 2). Part I a) Determine the Lagrangian for each system. DO NOT FIND THE EQUATIONS OF MOTION (5 points) b) What can you say about the systems based on the Lagrangian? (2 points) c) Solve for the equations of motion for both systems. (8 points) Part II Now, for scenario 1, introduce an additional coordinate and treat it as a nonholonomic system to determine the normal force acting on the mass. a)…

Chapter 9 Solutions

VECTOR MECHANICS FOR ENGINEERS: 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.

Find the mass moment of inertia with respect to y axis of the right circular cylinder by using direct integration.

Solution Summary: The author explains how to find the mass moment of inertia with respect to y axis of the right circular cylinder by using direct integration.

Concept explainers

Videos

Find the mass moment of inertia with respect to y axis of the right circular cylinder by using direct integration.

Want to see the full answer?

Chapter 9 Solutions