Two rods of different materials with coefficients of thermal expansion $α_{1}$ and $α_{2}$ respectively; Young's moduli $Y_{1}$ and $Y_{2}$ respectively; of initial lengths $l_{1}$ and $l_{2}$ respectively of equal cross-sectional areas are joined at one end. The free ends are fixed to two vertical walls as shown in figure.
Temperature of the surrounding is then increased by $T^{\circ} C$ . Find out the displacement of the joint of the rods.

Open in App

Solution

We know that
According to givencondition we have
$Y_{1} A \frac{(x + l_{1} α_{1} △ T)}{l_{1}} = \frac{Y_{2} A (l_{2} α_{2} △ T - x)}{l_{2}}$
$Y_{1} l_{2} (x + l_{1} α_{1} △ T) = Y_{2} l_{1} (l_{2} α_{2} △ T - α x)$
$Y_{1} l_{2} x + Y_{1} l_{2} l_{1} α_{1} △ T = Y_{2} l_{1} l_{2} α_{2} △ T - Y_{2} l_{1} x$
$x (Y_{1} l_{2} + Y_{2} l_{1}) = Y_{2} l_{1} l_{2} α_{2} △ T - Y_{1} l_{1} l_{2} α_{1} △ T$
On further solving we get
$x = l_{1} l_{2} (Y_{2} α_{2} - Y_{1} α_{1}) △ T / (Y_{1} l_{2} + Y_{2} l_{1})$ .

Suggest Corrections

Similar questions

Q. Two rods of different materials with coefficients of thermal conductivity 'alpha 1' and 'alpha 2' ; Young's modulii Y₁ and Y₂ respectively; of initial lengths l₁and l₂respectively of equal cross-sectional areas are joined end at one end.The free ends are fixed to two vertical walls. Tenperature of the surrounding is then increased by T°C. Find out the displacement of the joint of the rods.

Q. Two rods of different materials and identical cross sectional area, are joined face to face at one end and their free ends are fixed to the rigid walls. If the temperature of the surroundings is increased by

30^{o} C

, the magnitude of the displacement of the joint of the rod is (length of rods

l_{1} = l_{2} = 1

unit, ratio of their young's moduli

Y_{1} / Y_{2} = 2

, coefficient of linear expansion are

α_{1}

and

α_{2}

)

Two rods of different materials, having coefficients of linear expansion $α_{1}$ and $α_{2}$ and Young's moduli $Y_{1}$ and $Y_{2}$ respectively, are fixed between two rigid walls. The rods are heated to the same temperature. There is no bending of the rods. If $α_{1} : α_{2} = 2 : 3$ , the thermal stress developed in the two rods will be equal provided $Y_{1} : Y_{2}$ is equal to