Q. 29

# 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 be rod exert on one of the clamp? Area of cross section of the rad = 2.00 mm^{2}. Coefficient of linear expansion of steel = 12.0 × 10^{–6} °C^{–1} and Young’s modulus of steel = 2.00 × 10^{11} N m^{–1}.

Answer :

Given:

Temperature at which rod is under zero tension : T_{1} = 20 ° C.

Increased Temperature : T_{2} = 100 ° C.

Change in temperature : Δ T= T_{2} – T_{1} = 100-20 = 80 ° C.

Area of cross section of the rod : A = 2.00 mm^{2} = 2.00 × 10^{-6} m^{2}_{.}

Coefficient of linear expansion of steel : α = 12.0 × 10^{–6} °C^{–1}

Young’s modulus of steel: Y = 2.00 × 10^{11} N m^{–1}.**Formula used:**

We need to find the force on the clamps by the rod when the rod undergoes thermal expansion due to **increase** in temperature.

Formula for Linear Expansion:

Here, L’ is the changed length at T_{2} and L is the original length of the rod at T_{1}.

Thus, Change in length is

We get:

Formula for Young’s Modulus is:

Substituting Δ L:

Hence when the temperature is increased to 100° C , the rod will exert a force of 384N on one of the champ.

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