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.

Answer 1

If the temperature rises to 100°C, the rod on one of the clamps will exert a force of 384 N.

Explanation:

Zero tension rod temperature,$T_1$ = 20 °C

Final temperature, $T_2$ = 100 °C

temperature Change

Δθ = 80 °C

Cross-sectional rod field, $A = 2 mm^2 = 2 \times 10^{-6} m^2$

Linear expansion ratio for steel, $\alpha = 12 \times 10^{-6}$    $^{\circ}C^{-1}$

Young’s modulus of steel, $Y = 2 \times 10^{11} Nm^{-2}$

Let L be the steel rod length at 20 ° C, and let L ' be the steel rod length at 100 ° C.  

Change in rod length,

$\Delta L=L^{\prime}-L$

If F is the force the rod exerts on one of the clamps because the temperature rises, then

$\gamma=\frac{s t r e s s}{s t r a i n}=\left(\frac{F}{A}\right) \times\left(\frac{\Delta L}{L}\right)$

$\Delta L=\frac{F L}{A Y}$

$&L \alpha \Delta \theta=\frac{F L}{A Y}$

$&F=\alpha \Delta \theta A Y$

$=2 \times 10^{11} \times 2 \times 10^{-6} \times 12 \times 10^{-6} \times 80$

$=48 \times 80 \times 10^{-1}$

So, F=384 N

Therefore the rod on one of the clamps will exert a force of 384 N.  

Answer 2

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Force exerted by clamps = 384 N