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Question

A thin rod of negligible mass, length 0.5 m and area of cross-section 4×106 m2, suspended vertically from one end. The rod is cooled down and it can contract by 5×104 m but prevented from contracting by attaching a mass at the lower end. Choose the correct option(s).
(Given: Take young's modulus of rod Y=1011 N/m2, g=10 m/s2)

A
Value of mass is 80 kg
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B
Value of mass is 40 kg
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C
Energy stored in the rod is 0.1 J
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D
Energy stored in the rod is 0.2 J
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Solution

The correct options are
B Value of mass is 40 kg
C Energy stored in the rod is 0.1 J
Area of cross-section, A=4×106 m2
Length of rod, L=0.5 m
Possible contraction in rod, ΔL=5×104 m
According to question,


Force applied by attached mass (m),
F=mg=10m ....(1)
For Young's Modulus we can write,
Y=FLAΔL
F=YAΔLLF=1011×4×106×5×1040.5
F=400 N ....(2)
For the external load (F) on rod, equating Eq.(1) and (2):
10m=400
m=40 kg
Energy stored in the rod,
U=12×YAL×ΔL2
U=12×1011×4×1060.5×(5×104)2
U=0.1 J

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