Question

1
10. A composite rod is made by joining a copper
rod end to end, with a second rod of
different material but of the same area of
cross section. At 25°C, the composite rod
is 1 m long and the copper rod is 30cm long.
When heated to 125°C, the length of the
composite rod increases by 1.91 mm. When
the composite rod is prevented from
expanding by holding it between two rigid
walls, it is found that the constituent rods
have remained unchanged in length inspite
of rise of temperature. The Young's
modulus of the second rod is xx10%. Then
(x - y) is (Y of copper=1.3 x 101°N/m² and
a of copper=17x 10-6/K).​

Answer 1

Since total expansion = expansion of the constituent rods 1.91×10−3=0.3×17×10−6×(125−25)+0.7×a×(125−25)

or 1.91×10−3=5.1×10−4+70a

or 70a=(1.91−0.51)×10−3=1.4×10−3

When prevented from expanding thermal expansion is accompanied by elastic contration. Since the lengths remain unchanged, thermal expansion is equal to elastic contraction.

∴strain=taΔtI=aΔr=17×10−6×100=17×10−4

and strees=Y×strain=1.3×1010×17×10−4=22.1×106

For the second rod

Strain=aΔt=2×10−5×100=2×10−3

Stress =Y×strain=Y×2×10−3

But the same stress is effective throughout the composite rod.

∴Y×2×10−3=22.1×106 or Y=11.1×109N..m2