Question

A rod 1m long is 10 cm2 in area for a portion of its length and 5 cm2 in area for the remainder. The strain energy of the stepped bar is 40% that of a bar 10 cm2 in area 1m long under the same maximum stress. What is the length of the portion of 10 cm2 in area

Answer 1

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Answer 2

The length of the portion of 10 $cm^{2}$ is 0.55 m.

Explanation:

Given:

       S.E. in stepped bar is 40% of S.E. in rest of the bar.

       total length 'l' = 1 m

       Cross sectional area of stepped bar  = 10 $cm^{2}$

       Cross sectional area of rest of bar = 5 $cm^{2}$

    Let length of rest of bar be 'x'.

   Strain energy,

         ⇒ U = $\frac{\sigma^{2} }{2E} \times V$

        Now, compete both the strain energies of portion according to relation.

 

     ⇒ 0.4 x ($\frac{\sigma^{2} }{2E} \times x\times 10^{-3}$) = ($\frac{\sigma^{2} }{2E}\times 5(1 - x)\times 10^{-4}$)

   The maximum stress and the material of the rod are same. So they cancel each other.

    ⇒ 4x = 5(1 - x)

    ⇒ x = 0.55 m