A crystal has a coefficient of expansion $13\times10^{-7}$ in one direction and $231\times10^{-7}$ in every direction at right angles to it. Then the cubical coefficient of expansion is

The coefficients of thermal expansion of steel and a metal $X$ are respectively $12 × 10^{-6}$ and $2 × 10^{-6} per^o C$. At $40^o C$, the side of a cube of metal $X$ was measured using a steel vernier callipers. The reading was $100 \,\,mm$.Assuming that the calibration of the vernier was done at $0^o C$, then the actual length of the side of the cube at $0^o C$ will be

A metal rod of Young's modulus $Y$ and coefficient of thermal expansion $\alpha$ is held at its two ends such that its length remains invariant. If its temperature is raised by $t^{\circ} C$, the linear stress developed in it is

Give the value of coefficient of volume expansion at $0\,^oC$ for ideal gas.

An iron bar of length $10\, m$ is heated from $0°C$ to $100°C.$ If the coefficient of linear thermal expansion of iron is $ 10 \times 10^{-6}{°C^{-1}}$, the increase in the length of bar is  .......... $cm$ 

The coefficient of linear expansion of crystal in one direction is ${\alpha _1}$ and that in every direction perpendicular to it is ${\alpha _2}$. The coefficient of cubical expansion is