First slide

Elastic Modulie

Question

The composite rod ABC is acted upon by a tensile force F as shown in figure. Radius of AB is r and that of BC is 2r. If the total elastic potential energy stored in the rod is 100 Joule, the energy stored in the rod AB is $[Y_{AB} = Y and Y_{BC} = 2 Y]$

Moderate

A

84.25 J
B

75 J
C

68 J
D

88.89 J

Solution

Stress induced in AB is $σ = F / {πr}^{2}$
Stress induced in BC is $σ^{1} = \frac{F}{π {(2 r)}^{2}} = \frac{F}{4}$
$∴ U_{AB} = \frac{σ^{2}}{2 Y_{AB}} ({πr}^{2} . l) = \frac{F^{2}}{2 Y \times π^{2} r^{4}} \times ({πr}^{2} l) = \frac{F^{2} l}{2 {πr}^{2} . Y}$
and $U_{BC} = \frac{{(σ^{1})}^{2}}{2 Y_{BC}} [π {(2 r)}^{2} l]$
$= \frac{F^{2}}{2 (2 Y) \times {[π {(2 r)}^{2}]}^{2}} \times π {(2 r)}^{2} . l = \frac{F^{2} l}{16 {πr}^{2} Y} = \frac{U_{AB}}{8}$

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