Stress And Strain

Given,

%5Cfrac%7Bl_%7B1%7D%7D%7Bl_%7B2%7D%7D%20%3Da%2C%20%5Cfrac%7Br_%7B1%7D%7D%7Br_%7B2%7D%7D%20%3Db%2C%20%5Cfrac%7BY_%7B1%7D%7D%7BY_%7B2%7D%7D%20%3Dc

Free body diagram of the two blocks brass and steel are
Let Young's modulus of steel is

Y_1

and of brass is

Y_2

%5Ctherefore%20Y_%7B1%7D%20%3D%20%5Cfrac%7BF_%7B1%7D%20.l_%7B1%7D%7D%7BA_%7B1%7D%20.%5CDelta%20l_%7B1%7D%7D

.....(i)
and

Y_%7B2%7D%20%3D%20%5Cfrac%7BF_%7B2%7D%20.l_%7B2%7D%7D%7BA_%7B2%7D%20.%5CDelta%20l_%7B2%7D%7D

....(ii)
Dividing E (i) by (ii),

%5Cfrac%7BY_%7B1%7D%7D%7BY_%7B2%7D%7D%20%3D%20%5Cfrac%7B%5Cfrac%7BF_%7B1%7D%20.l_%7B1%7D%7D%7BA_%7B1%7D%20.%5CDelta%20l_%7B1%7D%7D%7D%7B%5Cfrac%7BF_%7B2%7D%20.%20l_%7B2%7D%7D%7BA_%7B2%7D%20.%5CDelta%20l_%7B2%7D%7D%7D

%5Cfrac%7BY_%7B1%7D%7D%7BY_%7B2%7D%7D%20%3D%20%5Cfrac%7BF_%7B1%7D%20.A_%7B2%7D%20.l_%7B1%7D%20.%5CDelta%20l_%7B2%7D%7D%7BF_%7B2%7D%20.A_%7B1%7D%20.l_%7B2%7D%20.%5CDelta%20l_%7B1%7D%7D%20%5C%2C%5C%2C%5C%2C...(iii)

Force on steel wire from free body diagram

T%20%3DF_%7B1%7D%20%3D%20%5Cleft(2g%5Cright)

newton
Force on brass wire from free body diagram

F_%7B2%7D%20%3DT'%20%3DT%20%2B%202g%20%3D%5Cleft(4g%5Cright)

newton
Now, putting the value of

F_1%2C%20F_2%2C

in E (iii), we get

%5Cfrac%7BY_%7B1%7D%7D%7BY_%7B2%7D%7D%20%3D%20%5Cleft(%5Cfrac%7B2g%7D%7B4g%7D%5Cright).%20%5Cleft(%5Cfrac%7B%5Cpi%20r_%7B2%7D%5E%7B2%7D%7D%7B%5Cpi%20r_%7B1%7D%5E%7B2%7D%7D%5Cright)%20.%20%5Cleft%5B%5Cfrac%7Bl_%7B1%7D%7D%7Bl_%7B2%7D%7D%5Cright%5D%20.%5Cleft(%5Cfrac%7B%5CDelta%20l_%7B2%7D%7D%7B%5CDelta%20l_%7B1%7D%7D%5Cright)

c%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cleft(%5Cfrac%7B1%7D%7Bb%5E%7B2%7D%7D%5Cright).%20a%5Cleft(%5Cfrac%7B%5CDelta%20l_%7B2%7D%7D%7B%5CDelta%20l_%7B1%7D%7D%5Cright)

%5Cfrac%7B%5CDelta%20l_%7B1%7D%7D%7B%5CDelta%20l_%7B2%7D%7D%20%3D%20%5Cleft(%5Cfrac%7Ba%7D%7B2b%5E%7B2%7Dc%7D%5Cright)

Chapter Questions
1 MCQs