# A round bar ABC of length 2L (see figure) rotates about an axis through the midpoint C with constant angular speed w (radians per second). The material of the bar has weight density y . (a) Derive a formula for the tensile stress a’ in the bar as a function of the distance x from the midpoint C. (b) What is the maximum tensile stress a m a x ?

### Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347

### Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347
Chapter 1, Problem 1.4.16P
Textbook Problem
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## A round bar ABC of length 2L (see figure) rotates about an axis through the midpoint C with constant angular speed w (radians per second). The material of the bar has weight density y. (a) Derive a formula for the tensile stress a’ in the bar as a function of the distance x from the midpoint C. (b) What is the maximum tensile stress a max?

(a)

To determine

Formula for tensile stress σx at distance x from C.

### Explanation of Solution

Given Information:

You have the following figure with all relevant information,

and weight density γ is given.

Calculation:

Consider the following free body diagram,

Calculate the mass of right hand section as,

Mx=γg×(Lx)×A      {where A is cross section area}                ....(1)

Centrifugal force is,

Fω=Mxω2×[x+(Lx)/2]=[γg×(Lx)×A]ω2×[(L+x)/2]=γ2g×(L2x2)×Aω2

(b)

To determine

Formula for maximum tensile stress σmax .

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