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Question

A 14.5 kg mass, fastened to the end of a steel wire of unstretched length 1.0 m, is whirled in a vertical circle with an angular velocity of 2 rev/s at the bottom of the circle. The cross-sectional area of the wire is 0.065 cm2. Calculate the elongation of the wire when the mass is at the lowest point of its path.

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Solution

Mass, m=14.5kg
Length of the steel wire, l=1.0m
Angular velocity, =2 rev/s=2×2π rad/s=12.56 rad/s
Cross-sectional area of the wire, A=0.065cm2=0.065×104m2
Let Δl be the elongation of the wire when the mass is at the lowest point of its path.
When the mass is placed at the position of the vertical circle, the total force on the mass is:
F=mg+mω2l
=14.5×9.8+14.5×1×(12.56)2
=2429.53 N
Young's modulus = Stress / Strain
Y=(F/A)/(Δl/l)
Δl=Fl/AY
Youngs modulus for steel =2×1011Pa
Δl=2429.53×1/(0.065×104×2×1011) = 1.87×103 m
Hence, the elongation of the wire is 1.87×103 m.

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