Question

Two steel rods and an aluminium rod of equal length l0 and equal cross-section are joined rigidly at their ends, as shown in the figure below. All the rods are in a state of zero tension at 0°C. Find the length of the system when the temperature is raised to θ. Coefficient of linear expansion of aluminium and steel are αa and αs, respectively. Young's modulus of aluminium is Ya and of steel is Ys.
Steel
Aluminium
Steel
Figure 23-E1

Answer 1

The length of the system when the temperature is raised to θ is  $\bold{5.8 \times 10^{4} \mathrm{Pa}}$

Explanation:

The steel ball Initial pressure = $1.0 \times 10^5$ Pa

The ball is heated to between 20 ° C and 120 ° C.

change in temp, Δθ = 100 °C.

steel  coefficient of linear expansion , $\alpha = 12 \times 10^{-6}$   $^{\circ}C^{-1}$

steel  Bulk modulus, $B = 1.6 \times 10^{11} Nm^{-2}$

we know that

$&P=B \times \gamma \Delta \theta$

$&P=B \times 3 \alpha \Delta \theta$

${\because \gamma=3 \alpha}$

${P=1.6 \times 10^{11} \times 3 \times 12 \times 10^{-6} \times(120-20)}$

${P=1.6 \times 3 \times 12 \times 10^{11} \times 10^{-6} \times 100}$

${=57.6 \times 107}$

$P=5.8 \times 108 P a$

So the pressure inside the ball is  $5.8 \times 10^{4} \mathrm{Pa}$