A rod of metal is clamped between two rigid supports and is heated by temperature θ. If wire is not allowed to bend then what type of stress will be set up in the wire and how does stress developed in wire depend on length of the wire?

Solution

When a rod of metal is heated, compressive stress will be set up in the rod. The wire is not allowed to bend and the force is applied normal to it's cross sectional area.
Let 'L' be the length of wire, A be the area of cross section, Y be Young's modulus and ' $α$ ' be the coefficient of linear expansion, heated to a temperature ' $θ$ '.
If the rod was free then increase in length of wire due to rise in the temperature will be L $α$ θ.

Here, since rod is not allowed to expand, therefore, compression in rod is equal to increase in the length of rod i.e. L $α$ θ.

If S is the compression stress in rod then,

$Y = S \frac{L}{Lαθ} = \frac{S}{αθ} or S = Yαθ$

Hence stress is independent of the length of the wire.

Some More Questions From Mechanical Properties of Fluids Chapter

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Mechanical Properties Of Fluids

A rod of metal is clamped between two rigid supports and is heated by temperature θ. If wire is not allowed to bend then what type of stress will be set up in the wire and how does stress developed in wire depend on length of the wire?

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