Question

A steel bar has a width of 10cm at 50°C. At what temperature will it fit exactly into a hole of constant width 10.005cm if coefficient of linear expansion of steel is 11*10^-6C-¹

Answer 1

By the definition of the linear thermal expansion we have:

$$\dfrac{\Delta L}{L} = \alpha \Delta T,$$ $$\dfrac{L - L_0}{L} = \alpha (T - T_0),$$

here, $\Delta L = L - L_0$ is the difference in the width of the steel bar after the change in the temperature, $L_0 = 0.1 m$ is the initial width of the steel bar at temperature $T_0 = 50^{\circ}C$, $L = 0.10005 m$ is the width of the steel bar after the change in the temperature, $\alpha$ is the coefficient of linear expansion for the steel bar, $\Delta T$ is the change in temperature.

Then, from this equation we can find the temperature at what the bar will fit exactly into a hole of constant width:

$$T = \dfrac{L - L_0}{\alpha L} + T_0 = \dfrac{0.10005 m - 0.1 m}{11 \cdot 10^{-6} \dfrac{1}{^{\circ}C} \cdot 0.10005 m} + 50^{\circ}C = 95.5^{\circ}C.$$

Answer:

$T = 95.5^{\circ}C.$