First slide

Elastic Modulie

Question

A rod of circular cross section has a length ‘l’ and radius ‘r’. It is subjected to a tensile force and the total elastic potential energy stored in the rod is U. If another rod of length 2l and radius 2r is subjected to same tensile force, Then elastic potential energy stored in the rod will be

Moderate

A

2U
B

$\frac{U}{4}$
C

$\frac{3 U}{4}$
D

$\frac{U}{2}$

Solution

(4)
For a rod of length ‘l’ and radius ‘r’,
$\begin{array}{l} s t r e s s σ = \frac{F}{π r^{2}} \\ ∴ u = E n e r g y d e n s i t y = \frac{σ^{2}}{2 γ} = \frac{F^{2}}{2 π^{2} r^{4} γ} \end{array}$
$∴ E n e r g y s t o r e d, U = π r^{2} l \times u = \frac{F^{2} l}{2 π r^{2} γ}$
Suppose total elastic potential energy stored in the second rod be U'.
$\begin{array}{l} ∴ \frac{U^{'}}{U} = \frac{l^{'}}{l} \times \frac{r^{2}}{{r^{'}}^{2}} = \frac{2 l}{l} \times \frac{r^{2}}{{(2 r)}^{2}} = \frac{1}{2} \\ ∴ U^{'} = \frac{U}{2} \end{array}$

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