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Question
A steel rod is rigidly clamped at its two ends. The rod is under zero tension at 20°C. If the temperature rises to 100°C, what force will the rod exert on one of the clamps? Area of cross-section of the rod is 2.00 mm2. Coefficient of linear expansion of steel is 12.0 × 10–6 °C–1 and Young's modulus of steel is 2.00 × 1011 Nm–2.
Solution
Given:
Temperature of the rod at zero tension, T1 = 20 °C
Final temperature, T2 = 100 °C
Change in temperature, Δθ =80°C
Cross-sectional area of the rod, A = 2 mm2 = 2 × 10-6m2
Coefficient of linear expansion of steel, α = 12 ×10–6 °C-1
Young's modulus of steel, Y = 2 × 1011 Nm-2
Let L be the length of the steel rod at 20 °C and L' be the length of steel rod at 100 °C.
Change of length of the rod, ΔL =L' - L
If F be the force exerted by the rod on one of the clamps due to rise in temperature, then
`"Y" = "stress"/"strain" = "F"/"A"/(triangle"L")/"L"`
`=> "F" = ("Y" xx triangle"L")/"L" xx "A" `
ΔL =LαΔθ
`=> "F" =( "Y" "L"αΔθ"A")/"L" `
F =YAαΔθ
= 2 × 1011× 2 × 10-6 × 12 × 10-6 × 80
=48 × 80 × 10-1
So, F = 384 N
Therefore, the rod will exert a force of 384 N on one of the clamps.
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